@Saint_Cad and others:
Let’s start with much simpler standards for distance and time units. Let’s base our distance units on the radius (suitably defined) of a ground-state hydrogen atom; and let’s base our time units on the oscillation period of radiation from the lowest level atomic transition of a hydrogen atom. We can return to messier standards in a bit. But for these standards, we have our basic distance unit being 1 “H radius” (symbol R, say) and our basic time unit being 1 “H period” (P). In such a unit system, a person might be 3.4x1010 R tall and might live for 6.2x1024 P.
If we measure the speed of light, we’ll find it to be around 2300 R/P.
You could then ask: why is it 2300 R/P and not 8 R/P or 99 million R/P? Could it have been something very different?
The reason I chose such basic unit standards here is that it showcases immediately how we are not, in fact, free to think about all three items independently (distance standard, time standard, and speed of light). To wit, we can “easily” calculate how the radius of the hydrogen atom depends on c, how the transition radiation’s period depends on c, and thus how their ratio depends on c. These dependences aren’t free to be whatever. The very structure of spacetime is part of the physics that underlies both standards.
Upon doing these calculations, one finds that the radius of the atom scales as 1/c and the radiation period scales as 1/c^2. Thus, it “turns out” (but it had to be) that the ratio of the radius to the period scales as c. If the speed of light were to suddenly halve, the size of a hydrogen atom would double and the period of the radiation would quadruple. Then, when we establish our distance standards and time standards in this modified universe, we would amazingly still find that the speed of light in this modified universe is 2300 R/P. It apparently didn’t change at all!
Saying the same thing in equations: the laws of physics relate these size and time standards according to 1~\mathrm{R/P} = \frac{3\alpha c}{16\pi}. Here, \alpha is the fine structure constant. No matter how you try to scale c, you will always find that the measurement of R/P scales by the same amount. That is, your distance and time standards will always scale suitably so that any re-scaling of c is unobservable. And, importantly, the numerical value 2300 reflects nothing about the speed of light and everything about the nuances of our chosen standards. In this case, the numerical value derives from a few mathematical constants and the fine structure constant. That is, 2300 is simply \frac{16\pi}{3\alpha}; it is unrelated to c entirely.
If you ask why is c equal to 3x108 m/s and not 87 m/s, what you are actually asking is “If I look at what I base my ‘meter’ on and what I base my ‘second’ on, what fundamental physics and corresponding emergent phenomena lead to those things having the ratio that they do?” That’s answerable, but it’s not the same as “Why does c have the numerical size that it does?” In fact, the “numerical size” question isn’t about c at all but about the physics that underlies (and relates!) the distance and time standards chosen.
For the hydrogen standards, the answer to “Why is c equal to 2300 R/P and not 87 R/P?” is directly visible in physics underlying the standards. If we try to do this instead for “rotation period of the earth” and “size of earth”, then there is a lot of smoke and mirrors in our path, but at the bedrock bottom of it all is the same story: the distance and time standards must be fundamentally related by the very spacetime they reside in.
The analogy of “horizontal” vs. “vertical” directions (say) being on equal footing can enter here. In the case of spacetime, the rule is just that “movement in a spatial direction can never be more than movement in the time direction”. When worded that way, there isn’t even a place to put a numerical value for c. It just doesn’t have one, fundamentally. Not any more or less than “up” versus “sideways” distances have numerical constants relating them. Any concept of c having a numerical value is, well, just down to units. Just like “2300” is unrelated to c but fully related to our unit standards and their physics.
A separate question might be “Why is c so big?” or more to the point: “Why is c so much bigger than speeds we experience in everyday life?” or more to the point still: “Why do we typically see things that move in the time direction much more than in spatial directions?” For everyday objects on earth, we need to have velocities commensurate with typical chemical binding energies and atomic masses so that it’s not all a dissociated mess every time two things touch. Similar lines of discussion, with different physics involved, could connect to gravitationally bound systems, say.
The point that I hope comes through the clearest in all this is that there isn’t even a place to put a numerical “input” value for c in physics. If you want to “change” it numerically in a given unit system, you need to change something else that changes how your distance and time standards relate, like \alpha (say) in the hydrogen example. “Changing” c does nothing, because there’s nothing to change. The law that “You can never move more in space than in time” doesn’t have a number associated with it. And this restriction is all that the speed of light amounts to.
(If I haven’t moved the needle here, consider this a starting point for further conversation.)