If it's dry now gaiters will do the same thing, I use one when in deep snow.
Up to a point, yes. In particular, good gaiters covers the laces, which otherwise collects a lot of snow (which will melt and try to find its way down, into the boot).
But gaiters don't protect the sides and front of the foot. If your boot isn't totally water proof you will get wet standing in deep snow. That happens a lot faster in wet snow than in dry snow.
The foot sack creates a totally snow free buffer around the boot. The difference this makes is amazing.
I guess by skiing boots you mean those meant for cross country?
Yes.