Why cut the whole hand?

[Gwaihir the Windlord]

Just studying my own wrist here, it actually becomes thinner after the hand, and then gets thicker again as you reach the forearm.
||||
||||
/||||\/
\||||/
||| Start of wrist
|| Continuation of wrist (exagerated)
||| End of wrist
|||| Start of Forearm

If the manacle was tightly around 'Continuation of wrist (exagerated)' then cuting off the thumb might not make any difference.

Also, Maedhros was starved and bony by that stage, so there wouldn't have been all that much soft tissue in his hand and wrist, although the same amount of bone. So we can change 'Continuation of wrist (exagerated)' to just 'Continuation of wrist', in this case, because the soft tissue filling out 'continuation of wrist (exagerated)' largely wouldn't be there.