How safe is it to reinstall macOS from Recovery, when backups are questionable?

In a pickle! How safe is it to reinstall macOS from Recovery, when there are issues with the Time Machine backup?

I'm on plan C or D here, involving problems with the system on the boot disk and problems with the external Time Machine disk, and hope that you all out there can help:

My mother has an Intel-based MacBook Pro, model 16,2 (13-inch, 2020) running macOS Tahoe 26.3. She has an M5 MacBook Air on pre-order, and I'm helping her prepare for the transition.

The fact that macOS 27 won't support Intel macs, and that she was getting relatively frequent application crashes, with the report saying "WindowServer experienced a problem" motivated the upgrade. (I have a screenshot with partial details, but that's tangential to the issues below, so I'm not uploading it here, though as I research, it may explain the sluggishness that also made her want a new machine.)

I told her to make sure to back up, and she told me she was getting errors that her TimeMachine backups, to an external hard drive, weren't completing.

So I did a screen share, and ran Disk Utility on the external drive, with this result:

Running First Aid on “Bottomless Pit” (disk2s2)

Checking file system and repairing if necessary and if possible.

Volume was successfully unmounted.

Performing fsck_hfs -fy -x /dev/rdisk2s2

Executing fsck_hfs (version hfs-704.60.4).

Checking Journaled HFS Plus volume.

The volume name is Bottomless Pit

Checking extents overflow file.

Checking catalog file.

The volume Bottomless Pit could not be verified completely.

File system check exit code is 8.

Restoring the original state found as mounted.

Problem -69842 occurred while restoring the original mount state.

File system verify or repair failed. : (-69845)

Operation failed…

It appears that Disk Utility put the external disk into a read-only state, then told me to back it up and reformat it. The disk only contains the Time Machine backups, and running Disk Utility and (per this thread) Drive DX on the disk rather than the volume suggested that it's most likely a corruption rather than hardware issue. So, I thought I would make sure the MacBook Pro's SSD is okay before reformatting the external disk and completing a fresh Time Machine backup. But the plot thickens...

We booted the MacBook Pro into Recovery Mode and ran Disk Utility.

• Checking the SSD at disk level gave a clean bill of health.
• Checking the Macintosh HD volume produced no errors either.
• Checking the Macintosh HD - Data volume scanned 15 snapshots and produced what seemed to be minor errors, and they are on the data volume, not the system volume anyway. (physical_size of dir-stats object is greater than expected, time machine [date code for various times today] inode resource fork xattr is missing or empty for compressed file, to name a couple)

However, after the repair completed and we went to reboot, we first got a dialog asking her to unlock Macintosh HD with her password, and then the dialog saying:

The version of macOS on the selected disk needs to be reinstalled.

Use Recovery to reinstall macOS or select another startup disk.

[Startup Disk...] [Recovery...]

I looked to see if somehow the startup disk wasn't properly selected, but the only option, of course, was Macintosh HD. I also tried booting in Safe Mode, but that gave the same dialog to push us into Recovery Mode. I ran Disk Utility again, with the same result (SSD and Macintosh HD volume okay, same errors on every snapshot on Macintosh HD - Data).

I tried to connect the external drive to another Mac to at least see from the folder names how current the last backup was, but it wouldn't mount there. (And, in fact, on that mac, Disk Utility wouldn't even show me a list of disks to scan... but that too is tangential.)

SO... I'd love advice on how to proceed from here...

1. Even though in theory it doesn't touch the Data volume, how much should I trust using Recovery to reinstall macOS, given that backups are questionable and possibly not current? Do I have no choice?
2. Is there another easy way to backup the MacBook Pro? Put it into Target Disk mode and at least copy the user folder to the other Mac? If in Target Disk mode, can I use the other Mac to back up the MacBook Pro to another drive with Time Machine?
3. Other ideas?

As if it weren't already well sub-optimal, note that I'm helping remotely from across the country via screen share and FaceTime.

(In case anyone asks: Documents and Desktop folders are not synced to iCloud. And, as a side note, we just a couple of days ago cancelled AppleCare+ on the six year-old MacBookPro because we were adding AppleCare+ to the new MacBook Air M5.)

Thanks in advance!!!!!