Why are electrical units (specifically, electrical current) considered a base unit? Note: this is NOT a question why current is the base unit as opposed to charge—that’s because measuring $1 \ \mathrm{ A }$ through a wire is easier to measure in a lab than is $1 \ \mathrm{ C }$ in free space; the question explores why electric units are chosen as base units in the SI in the first place. I am familiar with this question and have referred to it before. It does not answer my question.

Of course, according to Coulomb’s law, given equal base charges $q$, $F \propto \left. q^2 \middle/ r^2 \right.$.
For hypothetical purposes, consider a new unit of electrical charge—call it a $\mathrm{ \Xi }$ for fun.
Thus,
$$\begin{align}
1 \ \mathrm{ N } = 1 \ \left.\mathrm{ \mathrm{ kg }\!\cdot\!\mathrm{ m } }\middle/\mathrm{ s }^2\right.   &\propto 1 \ \left.\mathrm{ \Xi }^2\middle/\mathrm{ m }^2\right.  \\
1 \ \left.\mathrm{ \mathrm{ kg }\!\cdot\!\mathrm{ m }^3 }\middle/\mathrm{ s }^2\right. &\propto 1 \ \mathrm{ \Xi }^2 \\
1 \ \mathrm{ kg }^{ \left. 1 \middle/ 2 \right.  }\!\cdot\!\mathrm{ m }^{ \left. 3 \middle/ 2 \right.  }\!\cdot\!\mathrm{s}^{ -1 } &\propto 1 \ \mathrm{ \Xi }\\
\end{align}$$
It’s at this point that you can probably see why electrical units seem a bit less fundamental to me. Although the exponents aren’t integral numbers, a unit of electrical charge has still been expressed in terms of mass, length, and time, arguably the most fundamental units in our world.
In fact, as I understand, this is the dimensional form that the Gaussian unit statcoulomb aka franklin aka electrostatic unit of charge takes.
So why is an electrical unit in the base units of the SI if they can be defined in terms of mass, length and time? Why not define a unit of current that takes the form $\mathrm{ kg }^{ \left. 1 \middle/ 2 \right.  }\!\cdot\!\mathrm{ m }^{ \left. 3 \middle/ 2 \right.  }\!\cdot\!\mathrm{s}^{ -2 }\ $ instead of $\mathrm{A}$?
Also, in response to @Spirine’s answer, do systems of natural units (e.g., $\left.\mathrm{ MeV }\middle/ c^2 \right.$ from base $\mathrm{ eV }$) essentially have only one fundamental unit?
 A: To a large extent, what you're proposing is reasonable and doable. More precisely, the unit of charge you're describing

a xion of electrical charge, symbol $\Xi$, is the amount of electrical charge such that two charges of $1\:\Xi$ separated by $1\:\mathrm m$ will experience a repulsive Coulomb force of $1\:\mathrm N$

is pretty reasonable, and it is also remarkably similar to the definition of the ampere,

an ampere is the electric current which, when passed through two straight, parallel conductors set $1\:\mathrm{m}$ apart, will produce a magnetic force between them of $2\times 10^{-7}$ newtons per meter of length.

The only difference between the two is that in the former case (which is really just an MKS version of the statcoulomb) the Coulomb constant has been set to be truly dimensionless, whereas in the case of the SI ampere, we've set the proportionality constant $\mu_0$ to have a fixed value but with a nontrivial dimension.
In that sense, the ampere is exactly analogous to the (post-1983) meter: both can be obtained from a smaller set of base units (the second, for the meter, and the MKS triplet, for the ampere) in terms of a constant of nature ($c$ for the meter and $\mu_0$ for the ampere) which has a fixed value but a nontrivial dimension. That means, therefore, that the ampere is every bit as much of a 'base' unit as the meter is.

That bit of argument is, of course, a bit disingenuous, because when the ampere was defined science was many decades away from having a fixed value of the speed of light, but we did have a working MKS system with the meter and the kilogram defined in terms of the international prototypes, and the second set to a fixed submultiple of the solar day (before we realized that the Earth's rotation was too variable for accurate metrology). At the time, then, the MKS triplet of standards was as good as metrology got, and they were all very much independent, so your argument for fixing the dimensions of electrical charge was plenty valid - and indeed it was set into practice as the Electrostatic System of Units.
The problem, however, is that you can repeat exactly the same exercise as you've done in the question for the magnetic force between two conductors, and it provides some interesting contrast. Consider, therefore, the definition

a psion of charge, symbol $\Psi$, is the amount of charge such that if $1\:\Psi$ per second of charge flows down two straight parallel wires set a meter apart, they experience a force of one newton per unit length,

(i.e. essentially an MKS version of the biot). As you've done in your question, let's work out the relationship of our psion to the MKS triplet: since we're setting $F/L = I^2/d$, we have
\begin{align}
1\:\Psi^2/\mathrm{s}^2 & \propto 1\: \mathrm{N\:m/m} = 1\: \mathrm{N}\\
1\:\Psi^2 & \propto 1\: \mathrm{N\:s^2}=1\:\mathrm{kg\:m}\\
1\:\Psi & \propto 1\: \mathrm{N^{1/2}\:s\:m^{1/2}}=1\:\mathrm{kg^{1/2}\:m^{1/2}}.
\end{align}
So, everything is dandy - until we realize that we just got a unit of charge, $1\:\Psi$, which has physical dimensions that do not coincide with the dimensions of the xion you defined in your question. This is one of the big problems with the CGS systems of electrical units: the ESU and the EMU do not agree - not even on the basic physical dimensionality of electrical charge. 
This is, in many ways, a fundamental problem, because it means that one of either of Coulomb and Ampère's force laws is going to have a dimensional constant, or you're going to need to institute two parallel systems with duplicate units for everything. 
In some ways, the solution taken by the SI is "neither" to the above, by just striking out and deciding, for the sake of simplicity, that we're not going to examine the problem and that it's just easier to consider electrical quantities to have a physical dimension of their own. This immediately shuts down the issue, in a nicely symmetrical way, and as a plus side it lets you choose units which are of mostly real-world size.
A: Basically, you're asking why we should have different units to describe different measurements, since we could get rid of the dimension of proportionnality coefficients meant to have dimensionnally correct equations.
Taking your reasoning one step further leads to this. Let's assume that we have already eliminated A from SI standards units. Ideal gases law states that $PV \propto nT$; let's consider a new unit of temperature, called $\Phi$, then since $PV$ is energy,
$$ \rm 1\, J = 1\,kg\cdot m^2 \cdot s^{-2} \propto \Phi\cdot mol $$
So now we can replace $\rm K$ with $\rm kg\cdot m^2 \cdot s^{-2} \cdot mol^{-1}$. There are now only five base units, instead of seven!
You can keep doing this - that is, using arbitrary relations, getting rid of some coefficients that you consider useless, and saying that you can eliminate a SI fondamental unit - until there is only one unit left, ie. until units aren't used at all. And then, you will understand why there are (fondamental) units. Doing physics is studying and understanding reality. 
At the same time, we like to work with numbers, but a number has no link with reality: what means $1$? Is it 1 for speed, 1 for length, 1 for mass? Thus, we created units, which connect abstract numbers to the reality of the physical world, allowing physicist to understand what numbers mean. But there are a finite number of different quantities in the world, so we can use fundamental units. It appeared that many of them were linked: energy can be seen as the work of a force for example. Yet, it makes no sense to express the unit of something as fundamental as charge in matter of length, mass and time only.
A: Why are electrical units the base units in SI?   If I understand correctly, the major issue is the construction of benchtop standards for your precision metrology lab, then using them to adjust the most commonly-used lab instruments being brought in for traceable calibration.
Suppose you want to create an actual Standard Meter sitting on your lab bench, or a Standard Second, then use these to calibrate all of your other equipment.   The Base-unit physical devices need to be feasible to construct, and produce results with maximum digits of precision possible.  Yet also, if it's 1880, everyone is wanting to calibrate their quartz-fiber moving-coil mirror galvanometers, which give high-precision measurements of electric current.  That, and their microgram analytical balances in glass-and-cherrywood cases.  The Ampere and the Kilogram become base units, and 1840s French scientists create a standard Meter, which lets the high-precision calibrations spread all over the Victorian world.  (Heh, you then have to buy from Paris if you want a one-stage traceable calibration for your own Kilogram standard, and your own platinum Meter stick.)
No more galvanometers anymore, and today we can count individual electrons to obtain greater precision than by measuring seconds via atomic clock, and amperes by Kelvin balance.  Finally the SI standards are about to be revised, where physical constants such as Planck's and c and e become the Base for constructing macro-sized calibration devices.  Heh, once again charge becomes more fundamental than current.
WP: The coming revision of SI base units
SE ans: what is a 'base' in the new SI?
WP: history of the metric system
A: Charge/current do not have to be fundamental units. The SI system is also known as "MKS" for metre, kilogram, second. Before it was adopted, the CGS system was used.  CGS (centimetre, gram, second) is also a metric system, but in CGS the only fundamental units are the gram, centimetre and second. Everything else is defined in terms of them. Force, for example has the $\rm dyne = 1\:g\times cm/s^2$ as its basic unit, so $\rm 1 \:N= 10^5 \:dyne$.
For electric units, CGS has 2 systems, depending on whether you start with charge or current. In the electrostatic system (ESU), charge is defined by the force it exerts. The unit of charge, the Franklin (Fr), is the charge that, if 2 of them are 1 cm apart, exerts a force of 1 dyne between them. Hence charge has dimensions of $\rm mass^{1/2}\:length^{3/2}\:time^{−1}$, which is exactly what you propose in your question.
The unit of current is simply $\rm 1 \: Fr/s$. All other CGS units are defined in similar ways.
Similarly, in the electromagnetic system (EMU), where you start with current and the force it exerts, the unit of current, the Biot (Bi), is defined as the current which, if flowing in 2 parallel wires 1 cm apart, exerts a force of 2 dyne per cm. This means that the dimensionality of charge and current is different from that in the ESU system: charge is $\rm mass^{1/2}\:length^{1/2}$.
The Wikipedia article has a table showing the relationships between SI (MKS), ESU and EMU units.
