I don't really see how shoulder season pressure could be pushing bulls off public when all the hunting is on private. Kind of the opposite, really. It might be pushing them to other private where hunting is not allowed, but that's really nothing new. What I've seen the past couple years is the elk have changed their behavior somewhat. Rather than lollygagging around on private during the day (like they did a couple years ago when you actually had to draw a B tag to hunt the shoulder season in some places, before OTC, meaning much less pressure), they are now moving up onto public land during the day, but still coming down to feed on private during the night. Public land adjacent to private where shoulder season hunting has been allowed should be a good bet for opening day rifle at least, because the elk have been pressured to move onto the public to avoid shoulder season hunters. But I don't think they're going any farther up than they need to, at least not until the guns start going off on public.
The place where I've been hunting has also seen an influx of wolves and grizzlies in the past two years, and that's keeping the elk moving around a lot. Basically between the shoulder season hunting and the predators, the elk are always on the move, which again is really nothing new but I think it's even more so now, at least where I've been hunting. Life isn't as easy for them as it was a couple years ago. I can't say how much it has affected the population--I'm not sure anybody really knows that yet for sure. But I almost never see elk tracks without wolf tracks nearby anymore.
I also wonder about the increased popularity of archery hunting and how much that is going to affect the bull population. Seems like every joker goes out and arrows a six point these days. Just kidding, that's not true, but I don't think anybody can deny that the increasing popularity of archery hunting will have an affect on the bull population. Not to mention the increase in wounded animals never recovered.
We live in interesting times.