The idea that your muscles have enough strength on the upstroke to contribute meaningfully to cycling is actually a bit of a misunderstanding. Your leg muscles for lifting are much much much weaker than your leg muscles for pushing - they're designed to only lift up the weight of your leg and not much more.
The main advantage of the clips is to keep your leg in contact with the pedal throughout the cycle and make sure the ball of your foot is optimally placed on the pedal. With a traditional pedal you have to keep some downward force on the pedal to create friction to avoid slipping off, so even when that leg is rising up, you can't completely unload the pedal. With clips you can completely unload the pedal.
The main advantage of the clips is to keep your leg in contact with the pedal throughout the cycle and make sure the ball of your foot is optimally placed on the pedal. With a traditional pedal you have to keep some downward force on the pedal to create friction to avoid slipping off, so even when that leg is rising up, you can't completely unload the pedal. With clips you can completely unload the pedal.