Here's my take on fatter skis.
Many slides are propagated from a distance. This means there is a certain weight that will will overcome the tensile strength of a weak layer causing failure. So, the more weight, the more chance of propagating. So fat skis won't make much of a difference if you are propagating from remote trigger sites due to weight on a slope.
Another trigger is breaking through parts of a slab that were "holding" the slab from sliding on an already broken weak layer ie cutting loose a slab on a ski cut or sending something by skiing over a convexity. I think a couple inches of float extra is still splitting hairs unless you are getting those skis up to plane like a ski boat.
I wouldn't base my life or my buddies' lives on "couple weak layers here, good thing I brought the Birdos today".
But it goes back to all depends.
For the depth of concern, it really goes back to knowing the history of the snowpack. There have been some man eaters this season from the pnw over to MT that have 10 foot or more crown faces on some really weak layers. Most people only dig 6 feet or so. If you know of massive loading and some early season hoar, you might take the time to dig.
Trees-no way to be certain. Stuff slides in tight trees too. I like the addage open enough to ski, open enough to slide but it still depends on the snowpack, thermal affects of trees and sun, etc.
My .02 and answer to all of this is to follow general safety rules, know the snowpack and be humble.