What people want to see is "action" or what kind of danger is ahead of them. Many medic's run next or even infront of their escort, because they wanna help. In rather rare cases, they do - using the tazer or a flashbang. The problem however is that "usually" you just have a few zombies infront of you, making the medic think it's like that all the time, which tells him he can run past you or help out like usual.
But "sadly for the escorter" you also have special enemies and the medic runs in head first. If it's a soldier or a webber, it either hurts or slows you down. In both cases an ugly situation if you have currently someone down or highly damaged. You are forced to deal with it and may even need to patch yourself and your escort back up befor you can make your way to the damaged/downed person.
I still think seizing the enemies at a nest is still helpful, so the demo gets less damaged, the demo team has more "free" room (literally talking about the room or area the nest is in) to cover the demo and for the demo himself to take the given free time (by said seizer) to skip the part to fight your way to the nest and directly place the nuke. Since the seizer in most times has those few extra seconds available, him as additional defense help for the demo while he plants a nuke is surely a small but nice thing.
I'm starting to go way to deep into this topic, so I'll stop now.