Why does NASCAR use pushrod engines?

Answer 1

It helps keep speeds down.

A pushrod engine can only turn so many RPMs before the engine isn't able to properly control the valve train. It's called "float." When the valves float, you lose power.

An overhead-cam engine directly controls the valves, so engines set up thusly can turn far faster than pushrod engines. And all else equal, the faster your engine turns, the faster your car goes.