This is something of an aside on your first paragraph and lh712's. The theory of special relativity (and field theory in general) can be "derived" (as done in e.g., Landau, Lifshitz _Classical Theory Of Fields_) without empirical observation using concepts known to Galileo/Newton (if they count as "Medieval") - just with different assumptions/ideas about inertial frames. If you assume there is any phenomenon with a fixed observed speed in all "inertial frames" then you get special relativity with Einstein's gestalt-switch. After that it is, like so much in physics, a matter of thinking of an experiment to distinguish what matches capital-N Nature best.

Thinking otherwise mistakes a fundamental error that the only way for something to happen is how it historically happened. Reconciling observations forced relativity historically, but it could have been sussed out without those. That there were very fast but finite speed phenomena (which could motivate relativity reducing to Newtonian models) was seen in 1676 by Rømer with the speed of light and eclipses of Io, a Galilean moon of Jupiter. It probably could have been done with Galileo's telescopes in 1610. (That is just one example of a phenomenon with a very fast yet finite speed to suggest other experiments to test that "hypothetical relativity".)

TBH, this all relates to teaching "physics without calculus" and that sort of thing. Is the most clear presentation/derivation that which mirrors the history of our muddled yet ever demuddling ideas or that which starts from our most thoroughly demuddled ideas, best notations, etc.? The momentum is certainly the historical approach, yet there are notable exceptions.

Well, exactly, and I think that that is in line with what I originally wrote, and which is that you need the general relativity principle (by general I don't mean the general relativity but the concept of the equivalence of inertial frames) and the theory electromagnetism. (By the way, I am familiar with the Landau--Lifshitz textbook.)

(1) The relativity principle is exactly what you refer to yourself. You can't apply "Landau--Lifshitz"-like arguments without it. (And I don't think these principles count as medieval knowledge.)

(2) I mentioned electromagnetism, because you need some clue for the concept of an absolute speed that is same for all inertial observers. This is very counterintuitive from our everyday experience, and counterintuitive from the point of view of somebody living in the time of "Galilean" or Newtonian mechanics. Theory of electromagnetism is the only thing that I am aware of, that is nearly (by a stretch) accessible at the level of "medieval" knowledge, from which the concept of constant speed follows. (Historically: Maxwell's completion of previously inconsistent equations of electromagnetism yielded a wave solution propagating with the constant speed of light. This was interpreted in terms of the ether originally, but it is a strong hint in itself for Einsteinian principle of relativity; and a reasoning along the lines you alluded to can them be applied, at least in principle.)