Removing 7 old backups on Time Machine disk does rarely free up space

I understand you erased individual backup files, in an effort to increase available space. If that is correct, the Time Machine backup database is likely to have been corrupted and should no longer be considered reliable. Adding other, non-Time Machine files to a backup disk designated for Time Machine's exclusive use is not a good idea either.

The solution is to completely erase the backup drive and start a new set, for each Mac. You may use one backup drive to back up multiple Macs, but it's not ideal.

To learn how to use Time Machine, please read Back up your files with Time Machine on Mac - Apple Support. I recommend that you do not stray beyond the limitations of those instructions or expect Time Machine to do something other than what it explicitly states.

Very much agree with my More Learned Colleague on mucking around on the TM Backup Drive.

This is even more elevated if the TM Backup Drive was in the APFS Drive format expected for Big Sur and above.