This response is for Holybull, but it goes hand in hand with what you just said, and you're exactly on point.
Early in his career (in the 80's and early 90's) with the 49'ers, Steve Young would take off running at the first opportunity that he saw, and was though of more of a runner than a pure passer. This was a big source of frustration for players like Jerry Rice, John Taylor, and Brent Jones, since they were just on the verge of breaking things wide open right as he would take off scrambling. There were many a player clamoring for Steve Bono to be the full time starter, since the offense had worked almost as well as it did when Joe Montana were playing.
It really wasn't until 1994 that he finally learned to stay in the pocket, and be patient for the receivers to get open (which they usually did). He'd only take off if there were no other option, and his patience paid off in the form of an Super Bowl manhandling of the Chargers that year. That was arguably the year that he became more of a true QB with mobility, rather than a running QB.