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?
Answer
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.
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