Adding any new requirement would "break" the protocol. Changing the behavior of signature production/requiring a new field in the transaction indicating which input signed the txid, would make all old wallet/miners/clients/exchange software unoperative.
It would require forking the chain, because the blockchain does not actually contain that info. (I guess though the whole transaction signature might be dropped when the transaction is included in the chain; but then it would be only an "voluntary" rule enforced by miner to NOT include a mutated transaction by a third party.)
Anyway, such "breaking" behavior is generally undesirable.[
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There is always the possibility to create an alternative format and make it mandatory, starting with block 12345 some time in the future.
Obviously the clients would have to be able to process both the old format (before block 12345) and the new (block 12345 and beyond).
It is a fork, but it could be pre-announced well in time, if a large majority of miners agree.
Fundamentally, if the bitcoin algorithm could not be repaired and improved, it would die an early death.