I think that had Rare remained with Nintendo, they still would have declined, but would have released better games these past 14 years. And I believe so based on what known factors we have.
First, you have to remember Donkey Kong Racing. The game was announced prior to the GameCube's launch, and looked like a decent cartoonish racer, and seems to have been largely completed before the Microsoft deal went through. However, from what I can gather, Rare tried to rebuild it into a game called Sabreman Stampede. But that got cancelled because Microsoft wanted to focus efforts on Xbox 360 titles Kameo and Perfect Dark Zero. Not only was this a lost game, it shows how making several transitions in hardware must have been hell for Rare. Within five years, they went from developing N64 games (Conker in 2001) to the GameCube (SF Adventures in 2002), then the Xbox (Grabbed by Ghoulies in 2003), and then the Xbox 360 (Kameo and Perfect Dark Zero in 2005). Kameo was developed for all four consoles atvarious points. Had they stayed at Nintendo, that would have removed the GameCube to Xbox transition and would have possibly made the 7th gen leap simpler.
Grabbed by the Ghoulies showed the problems of Microsoft as a boss. It began development as a GameCube platformer in the style of Banjo Kazooie before being simplified on the Xbox. However, Microsoft pushed for the game to be released as a simpler and more linear experience. Hence, we got the critically panned final product. In contrast, Nintendo has been willing to give its 2nd parties a lot of leeway in development time. Rare experienced this during Conker's long development, and Silicon Knights had a similar amount of leeway with Eternal Darkness. So GbtG would have probably ended off better on the GameCube.
Last, I think Rare might have been able to find a niche with Perfect Dark had they stuck with Nintendo. Rare spent a lot of time working on that series before and after the Microsoft aquisition. You had Perfect Dark Zero, which was in development for the GameCube and Xbox before becoming a launch title for the Xbox 360, and got features cut to be ready for launch day. And you also had cancelled games like Perfect Dark Core. Problem was, Microsoft already had sci-fi shooters, Halo and Gears of War, and wouldn't approve of a new game after Zero. With Nintendo, Rare would likely have been free to pursue the series.