Why are electrical units (specifically, electrical current) considered a base unit?
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.
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.