Large amount of disk used by "System". Time machine not to blame. What's taking up so much space?
I don't think TimeMachine backups are stored elsewhere but on my machine, the Library folder takes up 100 GB! It may be included in the system size but OmniDiskSweeper separates it out. I've asked about the library folder being so large but didn't get an adequate response. I don't know how to safely reduce the size of that folder.

You should check the results of OmniDiskSweeper to see if your Library folder is taking up a lot of space. You have 2 library folders. Check the one in the Users/yourname directory.