I had a Big Sur MacMini update (non Beta) get stuck and nothing would fix it, so I then decided to go to Monterey (beta). After some initial problems, related to a Big Sur update waiting in the wings, the Monterey installation started - and then got stuck. The same problem: got so far then no further, time after time. I tried Safe Mode start-ups, but to no avail.
During a shutdown, which also seemed to be blocked, I became aware of HDD noise from the networked backup drive. A backup was in progress! When I turned off automatic backups in TimeMachine (and disconnected the drive), the Monterey install worked - and I suspect the Big Sur update would have done also.