I think it's reasonable for a DB like SQLite to delegate that to the filesystem. There is an overhead for doing it on the DB level, and since SQLite is just a file on the filesystem which, presumably, is serving many other files as well, why would you trust anything else on the filesystem if you don't trust SQLite? Like, your PHP script (or nginx server executable, or whatever) that is calling SQLite, that's not going to be check-summed either. Either you trust your filesystem or you don't, and if you don't, checksum and error correct on the filesystem level.

Though fair enough, it could offer it as an opt-in thing.

Because SQLite is a program, that will refuse to start or crash, and which you can trivially replace, if corrupted.

Whereas your sqlite data are your data, and if they're corrupted they can be lost forever or propagate the issue to backups.