At the very least, Mario Kart 9 (I mean a console entry, not Tour) won't be coming out for a long time. MK8D is the Switch's Mario Kart: while tradition can be broken, Mario Kart is a one-per console series. Hardcore fans might buy a MK9 on the Switch, but will the larger casual audience be interested, when they'd likely see it as 'but I already have Mario Kart for the Switch?' And that audience clearly matters, given how well MK8D sold
Besides, I'm not sure what a MK9 on the Switch could do to make it worth buying. It wouldn't look much better than 8 already does, it'd likely be a much smaller game than MK8D to start out with (32 tracks won't be looking so good when we got 48 tracks in Deluxe) and it'd need to have a significantly interesting gimmick to differentiate itself enough from 8.
RE: Mario Kart becoming more of a Nintendo Kart, that could happen, but I'm not sure how happy I'd be with that: I could see side characters in Mario that you don't otherwise get much of a chance to play as outside of Mario Kart being shafted for major Nintendo characters who while, more 'hype' we can already play as in their own series