How to completely turn off spell check in macOS Big Sur
It seems to be limited to the Messages app (so likely a bug) and after a few times of quitting and restarting (interestingly one restart was not enough) it too seems to finally have learned the lesson.
So my recommendation to others with similar issue:

Set the settings as you prefer
Don't forget about System Prefs / Keyboard / Text / Correct spelling automatically -  this is either new in Big Sur (not sure) or it was toggled on again after update
Patiently restart your apps a few times (or perhaps a full system restart) and this should resolve the issue