This idea of more actively promoting that questions asked on SO with certain game-development-related tags (unity, for example) be asked here instead is really something you'd want to bring up on the SO meta itself.
Game development questions aren't off-topic on StackOverflow though, so it's totally valid for those questions to be asked there.
Indeed there are quite a few questions on SO that appear to be game development related, but in fact are what we'd consider here to be game development adjacent (that is, the asker is making a game and describes a problem within that context, but it's fundamentally a basic programming issue and thus better suited to SO). I've occasionally poked through the game-related tags over there and not found a lot of questions I'd really be happy having here.
Automated tag-based suggestions can't pick up on those nuances and might very easily result in an unpleasant experience for askers, who start asking on SO, are suggested to instead post on GDSE, and then get their question immediately closed with the "ask this on SO" close reason.
It could also encourage cross-posting, which isn't cool.
I don't personally feel the need to campaign for this kind of thing; our community is different from the StackOverflow community, and we should respect their community and guidelines as they respect ours. There's definitely overlap in content sometimes but I think that's fine, I don't really think that each SE site needs to be so rigidly silo'd off.