Why cut the whole hand?

[Mister Underhill]

But Fingon could not release the hell-wrought bond upon his wrist, nor sever it, nor draw it from the stone.

I guess we may reckon that Morgoth put a little mojo on the “band of steel” around Maedhros’s wrist for added insurance against escape. Don’t forget that there was probably quite a bit of swelling around that wrist from weeks (months?) of hanging around. Also note that Fingon apparently first sought to gain entry to Angband, assuming Maedhros would be inside, and thus neglected to bring his surgical kit as the Squatter has already mentioned.