Why cut the whole hand?

[The Squatter of Amon Rûdh]

But you must admit that it would be better if he was very good with his left hand and good with his right rather than just have a left.

True. It would also have been nice for Maedhros if he hadn't been put on that cliff in the first place, but Dark Lords will be Dark Lords. Losing a hand is still better than the fate he'd chosen for himself, which is probably how Morgoth would have preferred him to escape.

BTW I think that Maedhros got off lightly: Imagine if he'd been held up by a band round his thigh; or his neck; or some far more personal part of his anatomy. That really would be nasty.