Unlikely, seeing as Rare's departure was around the time Nintendo acquired the remnants of the Turok boys at Iguana (Retro Studios) who would go on to become the "New Rare". The gamecube may have been a commercial flop, but the gamecube era was a time of great experimentation and freedom permitted by Nintendo, especially when it came to some of their most protected franchises. Mario had a paint brush, and Luigi had a vacuum cleaner. An effort was made to bring more mature content to the system with Eternal Darkness. As stated above Nintendo gave a new Texas-based studio the rights to bring Metroid into the 3rd dimension. Paper Mario TTYD had a noose in the main square and a Princess Peach shower scene. Mario Strikers allowed you to hip check mother fucking Toad into electricity and the character screamed in agony...
The Gamecube era allowed for more than enough maturity and creativity for Rare to absolutely dominate in, regardless of how the GC's low sales gimped every property. Unless they totally lost their mojo, an early 2000s Rare developing games for the GC would have been great to see. And had Rare survived those years, the Wii would have likely afforded them an opportunity to get a smash hit with a Wii Banjo-Kazooie game or maybe a Zapper game in the Perfect Dark universe. I think that would have been better than languishing in mediocrity for decades...
enjoy your discount (guano-shitting) Banjo substitute modern gamers!..... fuck Microsoft.