Time Machine says it doesn't have enough disk space. Can I safely remove really old backup files?

It should do that automatically.

I know you cannot manually remove anything but snapshots from the new APFS backups. But, I'm don't know if that is advisable.

I've been told by people that event the HFS+ backups cannot be manually removed anymore.

Does the message state it can't remove old backups to make enough space, or does it say that it cannot make a Snapshot? The latter message about snapshots is referring to the drive you are backing up, not the backup drive. You would need to clear some space from you startup drive in order to run a backup.