(My experience with broody birds is chickens) Why does she do that? Is she having trouble keeping all the eggs under her? Or does she not pluck her breast to expose her skin to the eggs for more warmth?

Yep, birds of prey usually also pluck a brood patch! Shes doing it to rotate and reposition the eggs under her to keep them all warm.

Best quality: her mama wiggles

I am no kind of bird expert, but I think it’s mama AND papa wiggles? At 0:06 and 0:41 that is definitely a different bird.

Yes, the gray headed bird is the male!