When, across any hardware divide, have we ever said "well, this still belongs to the previous generation"? That makes no sense.
There's very little about this rumor that has value, as is inherent to all "rumors," and all of this already constitutes more discussion than rumors warrant, in my estimation. But it would make no sense for us to be handed the first core Pokémon title on the Switch - if that's what this is - with the attendant upgrades and updates to everything and say "hm, no, this still belongs in the previous generation."
I would agree this rumor would have little value if it didn't have multiple sources backing it on top of the registered domains, little hints lending to its existence, and even Serebii reporting on it. This is not something I'd call a run-of-the-mill rumor, this has reason to be considered and discussed and it's completely fair to do both unless future information points to the contrary.
As for your second point, I think it will come down to how people are defining a generation. If someone defines it as a game within a new region, then this rumored game wouldn't be a new generation for them. However, if they defined it as something that cannot connect with the previous set of games and hypothetically this one doesn't, then they may consider this a new generation by that point. With that said, what makes a generation has always been fan-made and whatever this game is may just require us to be more flexible in our definition, we just have to wait and see what's going on with this release.