You all can downvote me if you want, I know this is just intended as a Trump-hate thread, and I hate him as much as anyone I know. But, after watching the video, here's what I saw.
He was very uncomfortable with that women's daughter on the left. Moved her away, then moved away from her, then shook everyone's hand except hers.
In all seriousness though, no one seemed to "pour out" and he wasn't frozen (he kept talking periodically on the mic and taking selfies/shaking hands). He also mentioned that they breached security in order to deal with the medical emergency. I'm guessing that with his heightened security protocols, he wasn't able to leave until they'd reestablished the outdoor perimeter. Very possible his motorcade also needed to move to allow for the ambulances, so those had to be brought back into position and then a security sweep had to be done again.
It appears that it's being explained to him on the left side of the stage at one point by secret service that he'll either need to rush out before the people that needed medical attention, or they'll need to breach security and he'll need to wait, and he clearly tells them to open it up and he'll wait. Probably could have been less awkward about the whole thing, but that seems to explain the last 45min of this thing.