This is a good point. 4E remains the only edition of D&D, and one a fairly limited number of TT RPGs to be concerned largely about the "fantasy" of being a class/being/etc., whereas videogames have been leaning that way for a long time. World of Warcraft's designers particularly clearly expressed this, and it's notable that one of the major changes of post-WoW MMORPGs which isn't simple mechanical emulation is taking a similar design approach. Indeed, FFXIV, which is probably now more successful than WoW, or close to it, is very devoted to this principle of design.Yeah, modern computer/video game design philosophy leans more into the playstyle of the class/archetype fantasy so that simply playing the game means that classes feel different in play. As you say, D&D (and its heartbreakers) tend to lean more into "fluff it up, buttercup" or "eh close enough."
I think a lot of tabletop RPGs have sort of tripped over this idea in the dark, benefited from it without fully understanding it, and then wandered off. White Wolf's oWoD games initially really seemed to get this. In the interviews in the documentary about them, Rein*Hagen seems to very much be expressing this idea. In Revised and later they drifted increasingly away from it, but the initial approach was very much about "the fantasy" of being the thing, which isn't just a "power fantasy", but also about how that power is expressed.
Other games, like Spire or some PtbA games seem to get this pretty well, but the industry default still seems to be "let's make a bunch of mechanics, then I guess we can cobble together some classes later" (or to not use classes or similar at all).