Motorized is motorized. If the trail isn’t open to motorcycles, it shouldn’t be open to e-bikes ESPECIALLY Class III which are essentially limited range motorcycles. If you can press a lever and not pedal at all, you have bypassed the biking part.
My wife and I got Class I e-bikes for camping with our travel trailer. We use them for transportation, and they allow us to go further and hit tougher hills than we can presently do on standard bikes given some medical issues. However, they are truly pedal assisted and you still have to work. We use them on forest roads and urban trails where legal and/or without power. They aren’t up to serious single track.
The Class III that don’t require pedaling are simply another category, and most of the dedicated “hunting” bikes I’ve seen are in that class.
Horses do raise a different question. Most, but not all, horse camps I’ve come across are commercial and they are getting those guys who couldn’t hike or bike back in to the animals. That’s buying your way in. Not saying that’s wrong, but this all gets complicated when drawing that line.
Are you only worthy of non-wilderness back country if you hike in? Is it okay to use the mechanical advantage of a bike? If you chafe enough, are horses okay? Other pack animals? I don’t find this to be straight forward, other than if you can press a button and go, you should not be able to go anywhere a motorcycle can’t.