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.)