I don't know for sure about "modern" vs... what, non-modern? Old? Primitive? What's the operating word for "not modern?" I think rather that RPG mechanics have done through waves of what was or wasn't faddish. Modern can be a question of perspective; is it newer than older stuff? Is it older than even newer stuff? Is 3e a modern game? Compared to what other games?
The earliest wave of design was very wargamish, obviously, and focused on tactical type combat. After that, there was a big wave of simulationist mechanics. Even as playstyle went through separate fads, I don't think the mechanics necessarily matched the playstyles super well. We now live in an era of splintered mechanics. 4e and 5e are heavily focused on tactical combat. Although 5e claims to be quick and easy to run and relatively light, it's not compared to other games. Pathfinder is even more gamist, focused heavily on character builds, optimization, and tactical combat. Are these not modern systems? PbtA have gone a completely different direction and have taken the indie game mechanics by storm, with tons of copycat games. This is a relatively "modern" conceit; the idea of narrative meta-mechanics and players forcing the game in a way that previously would have been the sovereign territory of GMs only.
The OSR, ironically given its name and genesis, has kind of devolved into simply rules-light indie games that may or may not even bear much resemblance to D&D specifically at all anymore; I'd say that most of the more recent games in this space are decidedly "modern" in spite of the purported "old school" nature of the playstyle that they're supposed to emulate.
I think the more interesting question, because it's actually answerable, is the last one of the OP; do I care if mechanics are modern or not? No, not in the least. Do I care that the mechanics are fit to function, i.e., do they accomplish what I need the specific mechanics to do better than other alternatives that may have been in place earlier? If they are, I don't care if they're new or old, only that they do what I want them to do. And in that case, my playstyle matters more than mechanics, or in other words, do the mechanics support my preferred playstyle? And that's a bit difficult to define maybe too; I often say that I'm old-fashioned but not old school, that I'm "paleo-trad"; kind of like regular trad play, but without the excesses and stumbles that people made in the late 70s and 80s in trying to define what trad meant, hopefully. My own game is broadly compatible with OSR principles, although I'm not really very aligned with the OSR as a playstyle, but I'm perfectly happy to add any mechanics that I like better as needed, hence I have minions, "action points" of a sort, advantage/disadvantage, etc. and many other "modern" mechanics grafted in. With a playstyle that I consider "Call of Cthulhu horror/investigation" and mechanics that I consider "a heavily revised rules light D&D variation" in a fantasy sword & sorcery-like setting.
But I don't consider my playstyle modern, since it's been more or less unchanged other than refinement of what works since the 1980s. Some of my mechanics are, admittedly, rather modernish innovations that once I saw, I was immediately struck with how well they would do compared to what I was doing previously.