Why don't we define temperature as a kind of energy? Currently, the temperature $T$ is defined in relation to the energy $E$ via Boltzmann's constant $k_B$ like so
$$ E \: [\mathrm{J}] = k_B \: [\mathrm{J/K}] \cdot T \: [\mathrm{K}]$$
I'm now proposing that you could redefine temperature as just another kind of energy. So, one Kelvin would be a unit of energy defined by
$$ T \: [\mathrm{K}] = \frac{E \: [\mathrm{J}]}{k_B \: [\mathrm{dimensionless}]} $$
Now temperature has the same dimension as energy.
My question
Would this redefinition break something? What would be wrong with defining temperature this way? Would it obfuscate some real difference between heat and temperature, which the current unit system preserves?
Some consequences of defining temperature this way

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*In the same way you can say $ 1 \:\rm kJ = 1000 \: J $, you could now write $ 25 \:\rm meV \approx 300\:K $, rather than $ k_B \cdot 300 \:\rm K \approx 25 \:meV $.

*The probability distribution in the canonical ensemble becomes $ \rho(E) \propto e^{-\frac{E}{T}}$, assuming $E$ and $T$ are both given in the same units, e.g. K, eV, J, etc.

*Entropy becomes a dimensionless quantity. Thus, entropy in information theory would be exactly equivalent to entropy in physics.

*Heat capacity would become a dimensionless quantity giving you the ratio of input energy to heat.

*Temperature would be made equivalent to heat. You could no longer distinguish between the concept of raising temperature and adding a certain type of energy, namely heat. They would be one and the same. (This last point is false.)

 A: Yes you can if you like define units of temperature in this way. Indeed physicists in their professional work are forever moving between things like energy, mass, frequency and temperature and employing energy units for all of them, or sometimes frequency units (such as when we say two energy levels in an atom are separated by 2 GHz or something like that).
In the case of temperature one should keep something in mind, however. Temperature is not quite the same sort of thing as energy. It is more a way of quantifying how energy is distributed. For example, for a system with a finite set of energy levels the mean energy in the high temperature limit is independent of temperature.
Also, you could have two systems whose energy levels have the same range, but differently spread out, and two such systems at the same temperature will usually have different mean thermal energy. So it is not quite right to equate temperature to mean energy as a concept. But it is legitimate to develop a system of units in which they have the same units.
Here are a couple more examples.
When we supply heat to a system, the temperature of the system might not change at all, or it might go up by a lot or a little. If you allow for the case of negative heat capacity (a property of some out-of-equilibrium states) then the temperature may even go down, if we allow that temperature can be assigned to some out-of-equilibrium states. An example occurs in a star held together by self-gravitation, where as the star emits heat its energy goes down but its temperature goes up.
Another thought. In heat transfer through a medium of finite conductivity, the temperature at one side, $T_1$, is not equal to the temperature at the other side $T_2$. When a given quantity $Q$ of energy flows through the medium in the form of heat, if it so happens that $Q = k_B T_1$ then clearly $Q \ne k_B T_2$.
A: Yes, the proposal of defining temperature as a kind of energy would break some important concepts rooted respectively in the concept of energy and of temperature. Here, I'll list some objective facts (just to show to people voting for closure as an opinion-based question that this is not the case).

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*Energy is usually an extensive quantity, while the temperature is intensive. It is true that it is always possible to get intensive energy per particle or per volume unit. But it is meaningless to multiply the temperature by the volume or the number of particles, to obtain a meaningful temperature.

*Energy is a conserved quantity while the temperature is not.

*By adding energy to a system, its energy content will increase. Its temperature not always (not in the presence of first-order phase transition).

*Temperature was invented to be a simple indicator of thermal equilibrium and of the direction of heat fluxes. This is not the case with energy (or energy per particle).

*Statistical Mechanics clearly separates the two concepts: Their relation is only through the dependence of the number of states on the energy. The appearance of the number of states, a third quantity not reducible to energy, should make clear the independence of temperature and energy concepts.

A: Temperature is a kind of energy just like time is a kind of distance. For some theoretical work, you can define $c=e= \hbar=k_T=1$ and end up with much simpler formulae. For most of physics and engineering, however, it is practical to have different units instead of expressing as dimensionless quantity.
A: Here is my standard example of how temperature and energy differ.
Put a sheet of aluminum foil in the oven and heat to maybe 400 F .  you can now grab that sheet; it'll briefly feel warm but won't hurt you. The temperature is high but the energy content (heat) is minimal.
Next, heat some water in a pot up to maybe  180 F . Stick your hand in.  Major pain from heat transfer results.  Temperature is low(er) but heat content is large.
A: Temperature is not energy. It is related to thermal energy of a macroscopic system but you still need to know the heat capacity, hence a lot of physics, of that system to arrive at its energy. Flawed proposal.
A: Would this redefinition break something?
As far as I can tell, it would not. You could define your unit system this way without creating any contradictions.
Is temperature a kind of energy?
Yes and no. It is not an energy in the same way heat is. If it were, it would make sense to say something like

We have a system with a total energy of 5kJ and will add 20K of energy to it

At first this makes sense. We have some system and its temperature increased by 20 K, thus increasing its total energy through the addition of heat, which is an energy. The problem is that the amount of energy we added is not equal to $20 \: [\rm K] \cdot k_B \: [\rm J / \rm K]$. The amount of heat added $Q$ would be given through the heat capacity $C$
$$ Q = C \cdot \Delta T $$
and since the heat capacity is a rather non trivial function, we can't simply get from temperature to heat.
However, in statistical physics, temperature is defined through the entropy $S$ and energy $E$ like so
$$ \frac{1}{T} = \frac{\partial S}{\partial E} $$
which means temperature is a change in energy per change in number of degrees of freedom and it has the dimension of energy per entropy. If you think about temperature this way it makes a lot of sense to make entropy dimensionless, as in in information theory, and think of temperature as a kind of energy. Note, though, that the sense in which temperature is an energy is very different to heat.
Temperature can be thought of as an energy in another sense. The temperature defines the energy scale at which the energy spectrum will be smeared because of random interactions with the surroundings. For example, let's say you wanted to measure two energies with an energy difference of 0.1 meV at 300 K. You would expect two peaks separated by 0.1 meV in your energy spectrum. However, the smearing in the peaks caused by random jostling and bumping will be on the order of $k_B T \approx \: \rm 25 meV$, meaning your two peaks will merge into one.
Would the redefinition obfuscate some real difference between heat and temperature, which the current unit system preserves?
All I can say is it made it easier for me to confuse heat and temperature, which are definitely different concepts.
A: If you redefine temperature as heat then you will need to introduce some new label to play the role that temperature previously played and you will need to incorporate conversation factors when performing calculations that involve the quantity formerly known as temperature (ie the quantity given the new label and measured by thermometers). For example, if I wanted to make coffee with water at what used to be known as the perfect temperature of 96 degrees, I would now have to refer to that quantity in a new way.
In short, you will end up with three units to work with (energy, temperature and the new label) where previously you had only two (energy and temperature).
Of course, you could streamline matters by dispensing with the redefined temperature, on the grounds that is it just energy after all, leaving you with energy and the new label.
And if you wanted to be really clever, you could adopt the word 'temperature' as the new label, which would avoid a huge amount of disruption all round.
A: Short answer

Why didn't we define temperature as a kind of energy?

Because it was impossible from a practical perspective. Temperature is the quantity you measure with thermometers, and we have only been able to create thermometers that directly measure thermal energies, to sufficient accuracy, in the past couple of decades.

Why don't we define temperature as a kind of energy?

Because it is now too late to change the unit system without an extreme level of disruption.

Long answer
Other answers have addressed the conceptual and thermodynamical aspects. However, there is also a much more mundane aspect: until recently, the redefinition you propose was simply not practically possible, because it was not feasible to measure temperature (to sufficient accuracy) using that definition.
Until recently, and since 1954, the kelvin was defined by fixing the triple point of water at $273.16\:\rm K$. This choice is distinctly practical: the triple point of water is a very specific temperature, and it is quite easy to create a cylinder with water and to identify when it reaches its triple point. This creates a physical system at the temperature that defines the kelvin, and you can then use this physical system as a primary metrological standard to calibrate any thermometer.
On the other hand, if you wanted to define the kelvin by fixing the value of the Boltzmann constant (either to unity, as you propose, or, as an alternative, to some arbitrary value), then you face a challenge when you want to create or calibrate a real-world thermometer. To do this, you would need, as the BIPM puts it,

a thermometer based on a well-understood physical system, for which the equation of state describing the relation between thermodynamic temperature $T$ and other independent quantities, such as the ideal gas law or Planck's equation, can be written down explicitly without unknown or significantly temperature-dependent constants.

In other words, your new "primary" thermometer needs to be a system for which you completely and thoroughly understand the thermodynamics from the ground up, with no layers of empirical or semi-empirical modelling. When the kelvin was defined, in 1954, this was an unattainable ideal.
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... but that's no longer the case. Over the past couple of decades, it has been possible to consolidate our understandings of the ideal gas law, and of the Planck radiation equation, into complete thermometers (respectively, acoustic gas thermometers and radiometric thermometers) which are as accurate, repeatable and reproducible as primary standards calibrated against the triple point of water.
And this is why, in 2019, the kelvin was redefined so that it is now based on a fixed value of the Boltzmann constant. (For a detailed look, see this previous answer of mine.) This redefinition, for all practical purposes, implements the change you proposed, retaining only a nominal layer of conceptual difference between temperature and energy. If we wanted to, it would be possible to reform the SI unit system to remove temperature as an independent base unit, and retain all of our existing metrological capabilities.
However, that change would be extremely disruptive, with very little clear gains from the change. Like it or not, the core structure of the SI (including the number and choice of base units) is by now fixed and too hard to change, even if several of the choices that we used to define it now make relatively little sense when compared against the reformed definitions. (For a similar, stronger case, see this discussion regarding the ampere.) These underpinnings are still reasonable enough that it's not fatal to keep them, so they'll be around for a long time.
