The more devices you have on at the same time, the more power you waste. On top of that, if one was pulling and the other was pushing at the same exact time, you would effectively get a short circuit.
The term "push-pull" refers to a circuit in which two transistors are used on alternating half-cycles to reproduce the input waveform at the output. Transistor Q1 conducts during the positive half-cycle while transistor Q2 conducts during the negative half-cycle. The two halves are then combined at the output.
