I updated a 2015 MBP this week, and have no fan issues. Big Sur's running fine, but getting it installed was painful! The download is 12.2 GB, after which 35-ish GB of free space is needed; few write-ups say so. Even after I moved the installer to a removable drive, the installer would eventually show I didn't have enough free space, without indicating how much or how much more, space was needed until I got fairly close to having enough.
After moving loads of files off of MacintoshHD, I used Manage Storage to see what else to target, and uninstalled a number of apps that I could later reinstall from the App Store (plus GarageBand sound libraries), and deleted thousands of email messages. The ultimate upgrade and everything since has gone great!
I could imagine extra activity to back up or sync new app versions and files, but I haven't yet restarted TimeMachine. While making room, I deleted some local backups, but the space they freed was negligible compared to what was needed.