What to do with people who aren't going to go away quietly
Stalling provides a hurdle that allows you to filter out requests.
If you put people on hold for six minutes, the trivial calls hang up. If you tell people that they can have something they've requested but will have to wait a long time, the unmotivated will go away. (I'm not proposing that this is a good way to handle customers, I'm merely saying that it does in fact triage the incoming requests.)
The question is, what do you do with the people who, from the start, are obviously not going to go away?
If a woman is in labor, you can try every demotivating tactic you can think of, she's still gonna have a baby. Might as well accept this and get her a room in the maternity ward, right now. Anything else is just annoying and a waste of time.
Don't try to talk a vegan into eating the chicken-fried steak just because the chef will yell at you if you ask for one more plate of steamed vegetables. It's not going to work, might as well skip the discussion and go get the veggies, with a smile.
It takes judgment to figure out who's not going to go away, but once you know, embrace the situation, don't struggle with it. It doesn't matter if it's not fair or not convenient. It is, it's here, it's now, and you win and they win when you accept it.