Description of the need While working on https://github.com/backdrop/backdrop-issues/issues/3979 I realized that the updater throws exceptions at an early stage of the process, when there's nothing broken (yet).
This might be confusing for admins, especially Backdrop novices, because an exception usually means, something's broken really badly. And that's not the case at that early stage of updating (files got downloaded and extracted, the install itself is untouched).

Doesn't this look like this Backdrop install is broken? It isn't.
Proposed solution If nothing's actually broken, it might be better to abort the update and show a helpful message. Log detailed information, possibly with links to documentation. We also need to consider that updating happens in batches. The fact that one project fails, doesn't mean all of them fail. (But that needs some more recherche...)
Alternatives that have been considered Sticking with UpdaterExeptions.
Recent comments
Thanks, the /tmp folder was the problem. I moved it elsewhere and then the updated worked. Thanks again!
Update 1.35.1 appears to install but hasn't
Probably not the reason, but worth to do a quick check: Sometimes the update cache interferes, so if something's odd with updates - some module (or core) claims to have updated, but then...
Update 1.35.1 appears to install but hasn't
Nope, not touched any permissions!
Update 1.35.1 appears to install but hasn't