Cross Check error with local Timemachine snapshot?

• Is there a way of fixing this problem with restoring?

Yes. Restoring a Time Machine backup erases the source volume prior to restoring its contents. As far as I know erasing the disk the only way to fix it.

• Given that these errors seem to be related to Timemachine snapshots, is it safe to restore from my Timemachine backups?

If you want to restore your data you have no choice in that regard.

You might find recommendations to delete Time Machine local snapshots. I do not recommend that because those alleged solutions only work in some cases, and even when they do work they just reappear again. More about that later (your last question).

• If I did restore, would I need to wipe the whole drive and the system files also, or just the data? (Disk utility gives no errors on the other (non-data) partition on the disk).

I would, but if you are sufficiently motivated, start with just data first and observe what happens.

• Could all this be a symptom of something worse? (e.g. failing HD)

Probably, but I wouldn't necessarily be concerned. They all fail—eventually—and as long as you have a Time Machine backup (preferably, more than just one) you can continue to use one that's operating in a state of failure until its performance becomes unbearable—frequent freezes and crashes you're no longer willing to tolerate.

The answer to those questions depends on your comfort level. If it helps though, my approach is such that if anything can go wrong, it will.

In other words: might erasing the source volume fail? Yes, absolutely. It might be so far gone that erasing it pushes it over the edge. Might the TM backup disk happen to fail during the restoration? Yes, absolutely, same reason. Might your Mac suffer a completely unrelated failure so that it won't even boot ever again? Yes. And so forth.

It's also possible that erasing the disk won't fix anything and you'll be no worse off than you are now.

Picture yourself in a position in which the workhorse Mac you depend upon for your livelihood essentially blows up, melts down, whatever, and in so doing takes out every single device connected to it and some that aren't. Your Mac is finished, unrepairable, done, and you have to buy another one. However unlikely it may be, does the prospect of that fill you with existential dread? If so reevaluate your contingency plan.

That pretty much sums up my approach, so if I were faced with the same circumstances I'd have no hesitation in erasing the Mac.

I have a backup of my main important files on Google Drive, but not another backup of Timemachine, so it wouldn't be a huge disaster, just a bit inconvenient in terms of productivity!

If that's the worst of it I think you're doing ok. Certainly better than most.

All those things have happened to me, and more. Lightning strike? Yes. Power failures? Sure. Arcing electrical faults? Check. Fires? Hah. Earthquake? It was minor, but yes. Global pandemic? So far so good.

No drone strikes yet but I'm ready. Bring it on.

Yet I have not lost a single solitary byte of data. I have work products dating back to 1984 that have been migrated to the Macs I use today.

It dismays me to read Discussions on this site describing a minor macOS update that didn't go well when the hapless user didn't implement so much as the thinnest most epithelial layer of defenses. You're doing fine.

Don't ask me about the upcoming coronal mass ejection event though 😬