Why can't "we" fix the quadratic hashing problem with OP_CHECKSIG with a soft fork?

Update NO_OPx to OP_CHECKSIG2 with soft fork, where OP_CHECKSIG2 doesn't quadratically hash (uses a single tx hash of all data for each input maybe?). Create a new transaction type for the new process like P2SH and after wide enough network adoption, soft-fork to invalidate any old OP_CHECKSIG transactions the way BIP 66 worked?

Or is there a fundamental problem with my assumption that "uses a single tx hash of all data for each input maybe" is possible at all?