Why cut the whole hand?

[The Squatter of Amon Rûdh]

What a delightfully gory discussion.
Actually, even cutting off the thumb as far as the wrist probably wouldn't make a hand fit through a sufficiently tight restraint: The hand's wider than the wrist on both sides, so as to accommodate all four fingers.

Mind you, I think Fingon was a little restricted by not having a set of surgical instruments to hand, so to speak. Bit difficult to do the fiddly stuff without a scalpel.

Perhaps somebody can remind me whether or not something was done to the cliff to stop it being eroded or chipped. Otherwise it's a job for a mason, not a surgeon.