You are correct that if you pull the mixture all the way back you’ll starve the engine of fuel, and it will stop producing power. However the airflow through the prop will still keep the prop turning just like wind turns a windmill. In order to have the prop stop turning, you need to either feather the blades to align them with the direction of flight (not something that you can usually do on piston singles) or reduce the airflow through the prop disk enough that the compression of the engine stops the prop.
The prop doesn’t just stop spinning when the engine stops (as long as there’s sufficient airflow)
I never really thought about it. So let’s say the engine stopped in flight because the mixture was leaned, if mixture rich would that send fuel and the windmilling prop would push the engine through the cycles thereby starting automatically without even using the ignition switch? Or ignition switch is required to create spark?