Would you need the *original* private key to spend the coins, or just any private key that ultimately hashes to the address?

You don't need the original private key to spend payments to a bitcoin address. Any private key with the same bitcoin address will work.

We can see this by looking at the script on a

random recent transaction. In order to spend the 50 BTC output of that transaction, we need to provide an input such that this script is satisfied:

OP_DUP OP_HASH160 f88b720031b65505f853bce809d4f4641744d2ae OP_EQUALVERIFY OP_CHECKSIGie. we need to put two values on the stack, one of which is a public key which has a hash160 of f88b720031b65505f853bce809d4f4641744d2ae, and the other of which is the signature obtained when signing the spending transaction with the corresponding private key.

At no point is there anything to distinguish the sender's private key - all we can see is the 160 bit hash of the sender's public key and so any suitable private key will do.