It doesn’t have to be local. In fact it shouldn’t be local anyway as backups accessible to be deleted from the source aren’t real backups any way.

You can the restore from a recent base backup and roll forward the WAL to just before the snafu.

Yeah, but that can still create a situation where you can create WALs faster than they're uploaded. (And if there's a good solution to that, I'm interested, because we ran a server out of disk this way last month at work)