It's just one of those things that don't make any sense whatsoever. Diglett can learn Aerial Ace, Shadow Claw, Rock Smash, Thief, and even Hone Claws (what claws does it have to hone...).
I guess if you really want a reason, then the fact that we don't know Diglett's body underground may account for it. Maybe Diglett has a hand or claws or something that is not seen that allows it to use Cut etc., but then that would raise the question as to why you, as the trainer, don't see what's underground if Diglett uses it to Cut trees.
Or maybe someone just tacked on Cut because Diglett doesn't learn a lot of TMs/ HMs.