I definitely think there is a cohesive idea behind them. I don't think we would quite be able to so easily identify a TTRPG instead of confusing it for a board game, or a war game, or a party game, or a parlor game.
Someone mentioned Fiasco, it's clearly a parlor game, but I've heard it described as a better introduction to RP than D&D....
I don't think a GNS style or Big Model style breakdown is useful or meaningful. Those feel like they're built to tell us a story about what TTRPGs ought to be. They're very is/ought oriented. Doesn't feel useful to me.

Interesting. I get a very different impression from them, and the Forge, generally. They seem to me like they're looking for answers about the relative popularity and success of the many TTRPGs that have come out over the decade. Like, why is D&D so successful? Well, clearly, simulationism is super-popular. But Champions is hella simulationist, how is it not pupular? Well, it's
purist for system, the popular simulationism is more about exploration, which D&D does well.
But, I also conclude it's not useful, since the success of TT games is primarily a marketing question.
But I don't know if you learn anything by trying to pin it down to one thing. It's kind of like... It simply feels like the best definition of a TTRPG is, "a game where the players are encouraged to roleplay that is played at a table," which is just tautological.
Nod. There's a lot of thought about the RP in TTRPGs, I guess since that's what sets them apart from other TT games. But RP is, itself, maybe not so easily defined? In usage outside the hobby, you'll find it in psychology, as an exercise to give you insight into someone else's PoV or feelings, and, of course, roleplay is totally a thing in The Scene, among swingers, and for vanilla couples just looking to spice things up....
But, the context of a TT game is pretty different from either of those.
I'd rather talk about the different kinds of subgames we collect together, and why the relationship between the sub games and minigames might help to reinforce or generate themes in gameplay and storytelling for some particular end. A more functional rather than categorical approach, because I don't think the categorical approach is working.
OK, what do you mean by subgames and a functional approach?
I keep coming back to it feeling like categorization as a means of gatekeeping what a de jure TTRPG is, and I'm not sure that's helpful or useful for creating the most diverse or interesting games.
There is definitely gatekeeping in the hobby, though the biggest perpetrator doesn't even do it on purpose (exactly), because there is a literal gateway to the hobby: the one TTRPG that managed to become a household name c1980, I think y'all may have heard of it. It's a de-facto gate-keeper, because, afterall if you try it and don't like it and never play a TTRPG again, well, you've been gatekept... not kept, literally, the opposite, the gate sent you packing.