You may not realize it, but with flat pedals, it's necessary to keep slight pressure on the rising pedal to keep you foot properly located. That means that one leg is slightly fighting the other leg.
With cleats (also called clipless), you can actually exert a lifting force on the rising pedal. It's really not very much at all, but it's better than pushing down.
Cleats may not work for you, but many riders, myself included, have literally ridden tens of thousands of miles without falling. I use SPD and have not trouble unclipping. I don't even think about it. It's become automatic.