Dad is 93, lives alone independently, and insists he's a competent driver. Even if he is OK, should we let him still drive short distances?

Some 93-year-olds can drive okay. If his doctor agrees he's competent, short distances should be okay. But you may want to check with the DMV. See if he needs testing. People age very differently, so age shouldn't be the only criteria. You can also check with the police.
Good luck. If he has to stop driving, doing it slowly is good when possible. So, do work on weaning him, anyway.
Carol