Disk First Aid

You can't fix a volume that you're currently booted into. You must be booted into another volume. Otherwise you're only verifying the integrity of the drive, not fixing it.

You must boot into the Recovery volume, i.e. boot with the Command + R keys held down  - Intel Macs or How to Boot an M1 Mac into Recovery Mode. 

Then rerun the First Aid until you don't get any error notification. It sometimes takes 2-3 tries.

I have two thoughts to share with you. First, when running Disk Aid it is sometimes necessary to re-run it multiple times to fix a problem. Second, if you do have to reinstall your operating system it will not affect any of your user installed files or apps. Reinstalling or upgrading an OS will also reinstall only those Apple apps which are part of the OS. This is assuming that you do NOT elect to erase and reformat the disk, in which case everything will be removed.

If you read the Disk utility report you showed us you can planely see that the error was NOT fully repaired. Sometimes it i necessary to run Disk Utility multiple times until you get a report that says the disks ok.

I don't know why it does that but it might be due to a problem with the TM drive. You cab try running First Aid on it from Disk Utility. You can do that for the standard boot.

It will take a much longer time to complete due to the complex ecosystem in the TM backups with hard links, etc. I let mine run over night.

As a last resort you can erase and reformat the TM drive to APFS and start over from scratch.

However, if you have any files on the TM drive that you don't have on your boot drive you may not 3ant to go down that road.

Duh! It says Data at the top. Missed it entirely. Was looking at the sidebar.

Try clicking on the other volumes and seeing if any of them have a list of snapshots at the bottom of the window.

The damage report came after the snapshots. So rerun First Aid from the Recovery volume until there's no report of corruption.

That's closer but no cigar. Get the window for the "- Date" volume and look at the bottom for the local snapshots.

If you have a lot of snapshots which make the First Aid process very long you can delete all of them except the latest one using Disk Utility like this:

Select all but the latest and click on the Delete (-) button. That will make the process go much, much faster.

You need to have the Macintosh - Data volume selected.

Glad you got the problem solved.

That's just the Details of the First Aid. You need to get a screenshot of the entire Disk Utility window like mine above.