Reddit posts collecting all of the below sources and specific quotes from them:
https://www.reddit.com/r/BanjoKazooie/comments/grmk7k/comprehensive_list_as_to_reasons_why_nuts_bolts/
https://www.reddit.com/r/BanjoKazooie/comments/gro2kc/child_post_list_of_comprehensive_quotes_from_rare/

GamesRadar 2008 interview revealing Nuts & Bolts has nothing to do with Microsoft's buyout:
https://www.gamesradar.com/banjo-kazooie-nuts-bolts-interview/

IGN 2012 interview revealing demographics mandate:
https://www.ign.com/articles/2012/08/22/ex-rare-member-shares-info-on-cancelled-game

Gregg Mayles 2018 interview revealing Rare wanted a different approach to platformers:
https://www.youtube.com/watch?v=9h9Y55vpHZo

VideoGamer 2008 interview revealing that they thought a platform-centric game would be boring to work on, and was not inspired by user-generated games like LittleBigPlanet:
https://www.videogamer.com/previews/banjo-kazooie-nuts-and-bolts-interview/

Wired 2008 interview revealing Rare thought platforming games stopped being innovative after the Nintendo 64, and the team expanded from 13 to 71 people so they could not make projects like they used to:
https://www.wired.com/2008/05/qa-banjo-kazooi/

Xbox Achievements 2008 interview revealing Rare thought the old movement system was stale, that they wanted players to come up with different solutions to problems, and assumed fans would be mad if they remade a game instead of a making a sequel:
https://www.xboxachievements.com/news/news-1567-x360a-Meets--Rare-and-Talks-Banjo-Kazooie--Nuts---Bolts.html

Gamereactor 2008 interview revealing Rare thought the blockier character designs felt more BK than the smooth versions of N64 models:
https://www.youtube.com/watch?v=rn9Il7IKgEI

The Rare Witch Project 2008 interview revealing Rare believed vehicle-building was the next evolution of platforming games, and that Nuts & Bolts was the third attempt at a third Banjo-Kazooie game:
https://web.archive.org/web/20081223105418/http://www.rarewitchproject.com/?id=1655

Kikizo 2008 interview revealing Rare wanted to test the power of the new Xbox 360 hardware:
http://archive.videogamesdaily.com/features/banjo-kazooie-nuts-and-bolts-rare-interview-p1.asp

Edge Magazine 2010 interview revealing Conker Live & Reloaded's negative fan reception caused Rare to rethink a remake:
https://web.archive.org/web/20101015012359/http://www.next-gen.biz/features/rare-vintage-part-two

Rare 2018 interview revealing Gregg Mayles was bored with the formula and did not know where to take it:
https://www.youtube.com/watch?v=p2QVnSJDAl4?t=3556

Steve Mayles 2020 interview stating Banjo-Threeie is Nuts & Bolts, putting work into reimagined worlds from Banjo-Kazooie is the same amount of work as new worlds, at one point Rare's management moved all but two people from the Banjo-Kazooie project, Xbox Market was different than the Nintendo market so they weren't sure it'd sell, and Rare was burnt out from making 3D platformers:
https://www.youtube.com/watch?v=Dp8n7QKJZHs