Hm. Yes, once a girl commits she will sleep there. I'm not certain eggs are required, as plenty of people report hens sitting on nothing at all when she's determined.
Here's what I'd suggest may be going on. You have a bunch of half broodies who are trying it on, but aren't all the way there yet. My girls do this off and on. They go from quiet in-and-out layers to screaming dinos who take longer, maybe even several hours, to lay. It does seem like a somewhat socially influenced behavior. But then they leave on their own eventually and act like normal chickens aside from a little fluffing and clucking.
If they commit, they generally stop laying and begin sitting in earnest, all day and night. I gauge by at least one solid night and day of sitting on fakes.
My main broody started first, and went all the way after a week of off and on symptoms. Several flockmates picked up the noises and the clucking from her, but never once spent a night and seemingly returned to normal. They did get a little worse after she hatched her babies, but again, it seemed to wear off.
If this is what is happening, all the crowding and fighting over that one box may be preventing anyone from getting serious. It still should probably be dealt with, but it's better than having 10 full on broodies in need of breaking!