As a payee, I don't care about verifying past ownership, I only care about
verifying the current owner which means I only care about this one
signature.
To validate the coin itself, I don't need to know it's history, I only
need proof that it ultimately originated from a genuine coinbase transaction,
which is also what the signature proves.
No, you've made a wrong assumption. That would be correct for account-based chains, but not for UTXO-based chain like bitcoin.
With UTXO chain, a full validation of transaction history is necessary to ensure that all of UTXO entries are valid. Including the one which was used for your incoming payment.
Otherwise it would be possible to inject fake coins through the protocol implementation issues.