Why didn't we replace our SI units with a better system? Intro
It seems to me that the SI units we use today are nothing but the result of a historical 'coincidence'.
I recently began researching about natural (absolute) systems of units, which are defined in such a way that selected universal physical constants are normalized to unity. These are very convenient, since almost all equations in physics are simplified. 
Planck Units
Take for example the Planck units, where five fundamental physical constants take on the numerical value of 1. 
To quote from Wikipedia:

Planck units have a profound significance since they elegantly simplify several recurring algebraic expressions of physical law by nondimensionalization. They are particularly relevant in research on unified theories.

Planck units are even dubbed "God's units", since Planck units are free of anthropocentric arbitrariness.
Justification
I know that many 'everyday' quantities for a physicist in terms of Planck units would be very small numbers (for instance) but that shouldn't be a problem if we use scientific notation. In fact, it's a great thing since it can give us a better picture about those quantities being now in a unit system which is conceptually linked at a fundamental physical level. Besides, many such 'everyday' quantities are also expressed by very small / very large numbers in SI units. Think Planck's constant.
Frank Wilczek even argues that using Planck units would help us re-frame important questions in physics:

We see that the question is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in [Planck] units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].

AI am aware that it would be a big upheaval if we are to change our SI units but if it is to be done it better be done right now before more physicists grow up using the old unit system. The change is not likely to render old written material useless since, for instance, Newton's Principia Mathematica remains a fundamental text in mechanics even though it uses outdated notions such as "quantity of motion" and "quantity of matter".
Question
Given all that I explained, is there a reason we choose to stick with the old and weary unit system?
Why didn't we leave the kilogram, defined by an arbitrary French rod, to ladies gossiping about Kim Kardashian's weight and choose something that makes more sense scientifically?
 A: Because it's a good thing that in physics we use the same units as in everyday life (at least, outside of a handful of countries which shall remain nameless). It makes things easier to explain, easier to measure. It makes it easier to check in your head whether the result of an experiment makes sense.
You're never going to get the whole world to change to Planck units, because the SI units are designed to be related to sizes we encounter every day. A meter is about half the height of a person; a kilogram can be easily lifted and is a good unit of measure for things like food; seconds, minutes, and hours subdivide the day into manageable chunks.
Suppose you use the Planck length instead of the meter. If you have kids, they will probably want to measure themselves every now and then. Are you going to tell them that they are $10^{35}$ Plancks tall? Or would you rather say that they were half a meter tall a few years ago and they are one meter tall now? This is just an example off the top of my head, of course.
The moral here is that SI units allow for easy numbers. You only have to memorize the prefixes milli- and kilo-, for example, to talk about distances as small as something you can barely see or as large as the size of the Earth, using no more than about four digits. This is simply not possible with Planck/natural/whatever units.
There's one caveat, of course: theoretical physics. This is the only place, among all the things everybody does in the world, the vast majority of which are not physics, where elegance of equations is important, as your quote about gravity says. But outside of that, in all the other subfields of physics, as well as for everyday life, your proposed units are useless at best.
A: Update: User User that is not a user comments that the New SI will be in use as of next week, 20 May 2019, five years after the posting of this answer.

A proposal to revise the SI is already being drafted (as we speak). It appears the 'arbitrary French rod' a.k.a. the international prototype kilogram (IPK) is already on its way out. BIPM's 'On the possible future revision of the SI' gives some of the details. (BIPM is not some obscure club, it is in fact the International Bureau of Weights and Measures. They also have the IPK in their basement.)

In the "New SI" four of the SI base units, namely the kilogram, the ampere, the kelvin and the mole, will be redefined in terms of invariants of nature; the new definitions will be based on fixed numerical values of the Planck constant ($h$), the elementary charge ($e$), the Boltzmann constant ($k$), and the Avogadro constant ($N_\text{A}$), respectively. Further, the definitions of all seven base units of the SI will also be uniformly expressed using the explicit-constant formulation, and specific mises en pratique will be drawn up to explain the realization of the definitions of each of the base units in a practical way.

The kilogram will get the following treatment. (This is a 16 December 2013 draft.)

The kilogram, symbol $\text{kg}$, is the SI unit of mass; its magnitude is set
by fixing the numerical value of the Planck constant to be exactly
$6.626\text{ }069\text{ }57\times 10^{−34}$ when it is expressed in the SI unit for action
$\text{J s} = \text{kg m}^2\text{ s}^{−1}$.


This is what the New SI will look like in terms of seven defining constants (again: draft).

The international system of units, the SI, is the system of units in which
the unperturbed ground state hyperfine splitting frequency of the caesium 133 atom $\Delta v(^{133}\text{Cs})_{\text{hfs}}$ is exactly $9\text{ }192\text{ }631\text{ }770$ hertz,
the speed of light in vacuum $c$ is exactly $299\text{ }792\text{ }458$ metre per second,
the Planck constant $h$ is exactly $6.626\text{ }069\text{ }57\times10^{-34}$ joule second,
the elementary charge $e$ is exactly $1.602\text{ }176\text{ }565\times10^{-19}$ coulomb,
the Boltzmann constant $k$ is exactly $1.380\text{ }648\text{ }8\times10^{-23}$ joule per kelvin,
the Avogadro constant $N_\text{A}$ is exactly $6.022\text{ }141\text{ }29\times10^{23}$ reciprocal mole,
the luminous efficacy $K_\text{cd}$ of monochromatic radiation of frequency $540\times 10^{12}$ hertz is exactly $683$ lumen per watt,
where the hertz, joule, coulomb, lumen, and watt, with unit symbols $\text{Hz}$, $\text{J}$, $\text{C}$, $\text{lm}$, and $\text{W}$, respectively, are related to the units second, metre, kilogram, ampere, kelvin, mole, and candela, with unit symbols $\text{s}$, $\text{m}$, $\text{kg}$, $\text{A}$, $\text{K}$, $\text{mol}$, and $\text{cd}$, respectively, according to the relations $\text{Hz} = \text{s}^{–1}$ (for periodic phenomena), $\text{J} = \text{kg m}^2\text{ s}^{–2}$, $\text{C} = \text{A s}$, $\text{lm} = \text{cd sr}$, and $\text{W} = \text{kg m}^2\text{ s}^{–3}$. The steradian, symbol $\text{sr}$, is the SI unit of solid angle and is a special name and symbol for the number $1$, so that $\text{sr} = \text{m}^2\text{ m}^{−2} = 1$.

Note that all are exact.

Example 1. What does that mean for a meter?
$$\text{m}=\frac{9\text{ }192\text{ }631\text{ }770}{299\text{ }792\text{ }458}\frac{c}{\Delta v(^{133}\text{Cs})_{\text{hfs}}}$$

Example 2. Kyle Kanos has a mass of $70\text{ kg}$ in the current SI. If he wants to make sure he has the "same" mass after the adoption of this New SI, he needs to make sure that his mass will be
$$70\cdot1.475\text{ }521\ldots \times10^{40}\frac{h~\Delta v(^{133}\text{Cs})_{\text{hfs}}}{c^2}=70\text{ kg}.$$
He can choose left or right. They are exactly the same in the New SI. (The $\ldots$ reflects numerical rounding of a quotient, not a measurement error or uncertainty.)

The BIPM has some FAQs about the New SI here.
A: The short answer is that it is simply not possible to design a "one size fits all" unit system. The staggeringly large range of mass, time and length scales that appear in the Universe prevent this. The Planck unit system you mentioned is mainly useful for people who will never touch an experimental apparatus. The vast majority of scientists and engineers do not even do physics, let alone theoretical quantum or cosmological physics, and need a standard unit system that reflects the magnitude of quantities that are most likely to be found in their everyday work. 
Thus, professional physicists use whatever unit system is most convenient for the problem at hand. There is no danger that "more physicists grow up using the old unit system", as if that would somehow obscure people's understanding. Actually the more redundant and bizarre unit systems that trainee physicists are exposed to, the better. This teaches you fluency in converting between these systems, and helps you to communicate with people from different subfields.
