Lyxen
Great Old One
That seems to be a strictly definitional problem. It sounds (to me at least) that your 'martial power' concept excludes the possibility of the sort of instant hit point recovery that we usually call 'healing'. I take it you're focused on the fiction not the mechanics as written (per your responses about shield).
Indeed, mechanically, 4e is more than fine, it's just that, in their quest for balance, they have I think diluted to much the spirit of what should be Martial Power for me (again, although not perfect, the 5e Battle Master is much closer to what I think) by forcing into it aspects which do no look martial at all, not only to me, but to the genre in general.
Given that your martial power concept excludes healing I think the argument cannot be advanced. You'd either have to add healing to your concept, or others would have to exclude it from theirs. Perhaps the focus should be on the question - why isn't healing in your martial power concept? Given that so far as the game mechanics are concerned, it seems to be.
See above, first it does not make Martial Power unique, it makes it too much of a clone of (for example) divine, second, it conflicts with its own self-declaration of being "not magic" when it produces effects which can only be magical.
Take Rocky. To me he seems like a pretty martial character. He several times rallies using... martial power? Or if not, why not?
Rallying oneself does not necessarily mean healing / recovering hit points. Once more, keeping on fighting for longer does not mean that you will recover better, whether in the short or the long term. It just tells you that you can keep on fighting longer. In that sense, the temporary hit points of 5e are a much better solution, narratively speaking, which has the added advantage of being specific to the martial power in their description, and the power is indeed called "rally".
Rally
On your turn, you can use a bonus action and expend one superiority die to bolster the resolve of one of your companions. When you do so, choose a friendly creature who can see or hear you. That creature gains temporary hit points equal to the superiority die roll + your Charisma modifier.
Close to perfect, IMHO, because it does not depend at all on the source of hit points of the target. If you take a meatbag, his hit points are probably mostly physical, if you take a lean fighter it's probably skill and resolve, if you take a slim rogue, it's probably mostly luck, and if you take an elegant paladin, it's probably divine favor. Healing hit points through magic does not cause any problem for all these cases, because it's, well, magic. Through self-recovery does not cause any problem either, the warrior renews his focus, the meatbag physically heals, the rogue "renews his store of luck" (see Mat Cauthon and the times he says that he probably used all his luck for a day), and the paladin prays. But stiffening one resolve should not replace all that, and indeed, it does not in fiction of the genre. It might allow the types above to fight longer (temp hit points of a different "source"), but there is no reason for it to replace lost blood/faith/luck through something that is not magic.
Once more, I'm not saying that it cannot be done, 4e is technically really good, it's just that, for me, it is not as suited to the narrative, for sure about the games that I run and play, but also, as demonstrated with now many examples, in the movies/books/shows of the genre.