You are raising a couple of great questions! There are a handful of extra (disjoint) pieces of information that I think are important to this conversation. You might be aware of some of them, but I will enumerate them just in case.

The "speed of light" c is much more fundamental than simply the speed of EM waves in vacuum. While humanity first learnt of this constant in classical EM, in modern physics its roots are much deeper. Here are a few equivalent "definitions" of c that do not involve EM:

- c is the speed at which massless particles travel in any reference frame (from special relativity)
- c is the maximal speed at which information can propagate (vaguer, but present in relativity, quantum, and QFT)
- if a given force fields (for whatever force) has the form ~charge/distance^2, then the carriers of that force field will propagate at the speed c (both EM and gravity)

As you can see, c is much more important than mere EM. Thus, it is important to start with c, not with μ0 and ε0, because c appears in parts of physics that do not know of μ0 and ε0 (e.g. theory of gravity).

But let us get back to μ0 and ε0. I will focus on ε0 first. This constant lets us relate an amount of charge to the force that charge will create. Historically, we have ended up with measurement style in which we measure force in Newtons and charge in Coulombs. But this is just an engineering prescription. In that particular prescription we need ε0 to have units and a specific clumsy value, because of the historical clumsy choice of Newtons and Coulombs.

You can see how Coulomb is indeed arbitrarily defined here https://en.wikipedia.org/wiki/Coulomb If instead we had defined a charge unit (or a current unit) from first principles, we would not have needed this extra ε0 to fix our silly choice of units. An example of a first-principles less-arbitrary choice of unit of charge is the Franklin https://en.wikipedia.org/wiki/Statcoulomb

Thus both ε0 and μ0 are present just because at some point in history we decided to use Coulombs to measure charge, etc. Similar to how Na, Avogadro's number, is just an arbitrary fixed number of atoms, that we use so that our textbooks do not have to say "use 10^20 atoms of this compound". Na is not a fundamental constant either.

However, there are systems of units in which ε0 and μ0 are simpler. The cgs system of units is a great example. Check out the values of ε0 and μ0 in that system https://en.wikipedia.org/wiki/Centimetre%E2%80%93gram%E2%80%...

Lastly, while we can dispense of ε0 and μ0 simply by picking a better system of units for our measuring devices, we can not do that for c and α. As already mentioned, "c" is the speed at which information propagates (which is important for fundamental notions like locality and causality), while α is related to the ratio between fundamental forces.

By the way, in cgs units α does not contain μ0.

In conclusion: at the end you have two reasons to say that c and α are more fundamental than ε0 and μ0:

- you can not remove c and α from your equations simply by changing the units of measurement (well, you can set c=1, but this is another can of worms unrelated to this discussion)
- c appears in fields unrelated to EM, while ε0 and μ0 are only in EM