Only reason I have my doubts about Switch's successor console happening earlier than late 2023 is because I feel like Nintendo will want to position a new Mario Kart game within its launch window as their big sales driver, and it's hard to imagine that while DLC for Kart 8 is still ongoing. MK and 3D Mario are imo the big things they'd be priming up.
But I guess we'll see what happens! It doesn't feel like we're working with traditional console cycles, and the next Switch is bound to be backwards compatible anyway.
That's one of the strongest argument to believe that the next console is either late 2023 or 2024.
The next Mario Kart should in my opinion absolutely release within the first 12 or even 6 months of the new hardware; it helps the console selling, and it just doesn't stop on selling throughout the entire generation. As you said, it's doubtful that a new Mario kart would release before 2024, even though I'm willing to bet that the development of the game is probably well underway.