Differences between starting a road bike vs a mountain bike?
I ride a mountain bike and my feet have no issue hitting the ground and waiting 
  until it's time to go again. That's not the case with the road bike, where I
  have to get out of the saddle when stopped.


You say you have to get out of the saddle when you stop the road bike. Is there a large difference is saddle height between your road and mountain bike? Are you staying sat on the saddle with a foot on the ground when stopping the MTB, but have to drop out of the saddle to get a foot down when stopping the road bike?

You should be able to reach the ground with your toes when sitting in the road bike saddle, without tilting the bike. You should be able to start sat in the saddle, one foot on the ground and the other on a pedal. Make sure you stop in an appropriate gear so  pushing down on the pedal gets you underway and able to lift the other foot of the ground.

Look at the saddle height on your road bike. It may be too high. Also your MTB saddle may be too low.