No, what would cause you to give up the lead is almost always the inability to stop the opponent from passing well. If you stop the opponent from passing well, you're unlikely to give up the lead.
And the idea that a great running game is just going to be able to "run out the clock" every time is a pipe dream. It almost never happens in the NFL. At some point, you're going to have to pass successfully if you want to pick up a first down and keep the clock running. In all of 2014, with that vaunted running game, we had only one possession that lasted more than four plays without having to pass when we led in the fourth quarter -- and on that possession, we STILL had to punt because we ran on third-and-5 (we had a 16-point lead against one of the worst teams in the league, so it didn't hurt us). We never had another possession without passing that lasted more than four plays or took more than 2:28 off the clock.