I already know that the run header resets the Pi. My question is does one of the pins lead directly to ground? I would like to use a common ground on the GPIO pins and connect it to the run pin 1.
Is this doable/safe?
You don't say which Pi model you mean. However, I assume (but have not checked) that all run header implementations are the same.
On the Pi 3 Model B one of the run header pins is connected to ground.
All Pi grounds are connected so it does not matter which ground you use.