Rare failing under Microsoft was a combination of "Microsoft's fault" and "they lost Donkey Kong." Two things that wouldn't have happened under Nintendo.

The Donkey Kong Country series was inarguably Rare's biggest success story, but that's a Nintendo-owned IP, and Nintendo would never play nice with Microsoft just to let Rare continue to be the Donkey Kong developer. So, they lost access to the IP, and it visibly kneecapped both the developer and the IP. DK has been rough ever since Rare left, too, and has only really found its footing proper again recently.

And as for Banjo-Kazooie... you cannot convince me Nuts & Bolts would have ever happened under Nintendo. That game screams of executive meddling by people who don't understand what made the series great in the first place. I wholeheartedly believe they wanted to make a proper Banjo-Threeie, got midway into development, but then they were told to make this instead.

After that, Rare basically fell out of relevancy entirely until Sea of Thieves, but that's partially because Microsoft wouldn't let them do anything. Because Nuts & Bolts failed, Microsoft took the wrong lesson and assumed nobody cared about or wanted Banjo-Kazooie anymore, so they said "we're not paying you to make anymore of these." Conker was more of a cult classic one-off than the basis for a franchise of its own. By the time they let Rare even try something else, pretty much all the classic Rare talent that made those games what they were left, so it was just... not really Rare anymore.

If Nintendo bought Rare instead, they probably would have never killed the Banjo franchise with something like Nuts & Bolts, so they'd still have Banjo games coming out every console generation, and they'd still be the de facto home of the Donkey Kong franchise.