If the circuit would be operated in the described way, there's still a signal path through M1, M3 and M4 which contributes half of the output signal. In so far it's not correct to say that M1 is "just biased". M1 and M2 form a differential pair, if M1 current is increased by a positive gate signal, M2 current is reduced by the same amount.
Another point to consider, a single stage amplifier can't be seriously used as OP, it neither has the output swing nor the required gain. It can be operated as inverted amplifier with a very limited voltage range.