Big Sur 11.6.5 now working on my Macbook Pro late-2013. The procedure outlined by @sf98723 worked for my 11.1 upgrade, but not 11.6.5. Similar procedure that worked for me - https://forums.macrumors.com/threads/big-sur-and-applethunderboltnhi-kext.2267818/page-4?post=30708345#post-30708345. Key items:
1. Look for "rebuilding" output from this command:
bash-3.2# kmutil install -u --force --volume-root /Volumes/Mac*HD
checking collections...
Warning: com.apple.driver.KextExcludeList was not found!
considering release out-of-date because of --force
rebuilding release collections: boot, system
rebuilding release collections:
boot kernel collection
system kext collection
rebuilding local auxiliary collection
2. Added this step missing from original procedure:
$ kextcache -i /
Post #64 (open core legacy patcher) from the above link may be easier, but I read it too late.