Whenever there are gantries, wires and such gubbins above and around people these days, helmets are supposed to be worn in case of equipment becoming loose etc and falling on them.
A tether does not stop someone who suffers from vertigo from feeling great fear from being at an uncomfortable height.
I went in a cable car in Grenoble once and spent the whole trip clinging to the central pillar in it, I was convinced the floor was going to give way and that would be it. Very silly really but that was how I felt.
If any of them were really, really scared they couldn't have done it, tether or not.