The issue is more subtle than that, and the way you've phrased it is technically wrong. There are "reference frames" you can create in which light, or photons, travel with a different speed. This came up here:
(hmm, can't find link atm)
referencing the case of thinking about the viewpoint of a spinning person - the stars at great distances will seem to move "faster than light" thanks to $v = r\omega$, and so too will the photons, i.e. "light speed" will be higher, in fact, at suitable distance, arbitrarily high. (E.g. if you spin at an $\omega$ of 1 rad/s, and $r$ is even $1\ \mathrm{Pm}$, i.e. only 1/40th the distance to the nearest star away from the Sun, already the $v$ is $1\ \mathrm{Pm}/s$, far in excess of $c$, i.e. $3 \times 10^{-7}\ \mathrm{Pm/s}$ on this scale.)
In special relativity it's not brought up, but it is crucial in general relativity, and since general relativity includes special relativity as a special case (hence the name), the same cosiderations, technically speaking, apply to it.
Relativity is really a theory of space and time, and as said, it "requires the language of events, not things" (ref). More particularly, relativity is a theory about the laws that govern flows of information throughout the Universe. In its purest form, we are really only concerned with one kind of question, and it is this:
"Can you send a message from event $A$ to event $B$?"
"Events" are just points in space-time, to which we attach a signifier. The answer to this question is either "yes" or "no", a binary answer. For every pair of events in space-time, we can ask such a question, and the theory of relativity provides a mathematical framework that describes when the answer is "yes" and when the answer is "no". It also lets us work out how things look from the viewpoint of being inside a universe where the information we receive is subject to these constraints, i.e. what we can and cannot gather from the information coming at us at the little points in space-time we occupy. All the "weirdness" of relativity traces to basically this. Special relativity describes the form of those relations in the absence of matter, while general relativity describes how they are altered by the presence of matter.
Coordinate systems, or "reference frames", are simply ways to label events. What labels you put on them do not change the relationships between them. If I label the inside of my house "cooties", and the outside "znabby", that doesn't change the basic relationship of interiority/exteriority that exists between them any more than if I label them "inside" and "outside", respectively. (Same if I decide to confusingly call the outside "inside" and the inside "outside".)
What the Michelson-Morley experiment shows is not directly a statement about what happens in reference frames, or a statement about "aether", even - it is entirely logically possible to imagine a Minkowskian space-time filled up with an aetheric medium just as one can imagine a Galilean one so filled. Rather, it is a demonstration that the behavior of communications - of messages - obeys the former set of flow rules, not the latter.
And those rules can essentially be described as saying there exist a class of reference frames (coordinates, labels) that you can put on events, such that the permissibility of communication takes the form of a speed limit, and the transformation between these frames leaves said speed limit fixed. The reference frames follow from the limits, not that the limits follow from the reference frames. And one more result of the experiment is that it shows us that light, specifically, is a real-life medium of communication that saturates the Universal speed limit (to at least the experiment's error bounds, of course).
When you relabel those with something else, like a rotating reference frame, of course these relationships become harder to describe mathematically, but they are still the same in that in both frames you will note that communication between the same sets of events is or isn't impossible. E.g. while light may be going "faster than light" in the rotating frame, you won't see any ships making a trip from Earth to Proxima b in less time than 4.3 years (barring of course potential things like wormholes that require GR and even more, still-unknown post-GR physics to fully treat and also to assess the (im)possibility of).
Now, as to why the communication rules in our Universe take this form, there really isn't an answer, at least that you can have in physics and to the best of our knowledge. The only way you can answer "why" in physics is if you can derive it, as you are suggesting, from a more fundamental principle, and your circular argument shows you can't, and moreover, when phrased as above it seems pretty damn fundamental already, so I would doubt that we will ever find such a reason. You have to start somewhere.
The best way to convey it is to just say that that our totality of empirical observation has been consistent with the idea that the Universe has a speed limit, and no exceptions have been found. That's it; it's "how it was made".
