You should always handle exceptions if you know what they are about, and crash if you don't.

Players overwhelmingly prefer a crash to a corrupted save.

Handling the exception doesn't mean you have to fix the bug - writing to the log and implementing a "temporary" workaround is perfectly fine, if you decide that the bug is acceptable. But to decide if a bug is acceptable you have to first know what it is.

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> it's really hard or impossible to predict most situations when this could happen.

If there are lots of unhandled exceptions that are impossible to predict, there are major flaws in the software that must be fixed. If there are few unhandled exceptions, you find them with Unit Testing, then internal testing, then alpha and beta testing.