I tend towards enduro more than motocross - I'm not that great at jumping sadly.
And it was from this I was thinking along those lines - I've ridden enough cross country events with ICE powered and foot powered to see the difference the machines get put through.
One of the reasons I love enduro (unfortunately my wrists don't, so not done much recently

) is that it's like riding the fun down hill bits of a push bike cross country, but faster and the same speeds uphill too.
I'm surprised you mention snow riding. I have a bit of experience with that - I've used various bikes in the snow from a GSXR1000 to a KTM 690 with studded tyres for hard-pack road riding (can't be doing with the traffic we get here when we get snow).
Thin tyres don't do well in my experience. This isn't mine, but I recently got one of these with the intention of using it for snow fun if we get any this year - https://www.youtube.com/watch?v=FtaD-LKwIgg - big wide wheels are what is needed from what I've seen. Ideally I want to replace the front wheel with a second rear and add an electric motor above the front wheel to make it two wheel drive. But many other 'projects' first.
I recently got some 17/14 rims for a (normal, not fat tyred) pit bike I've got and while the tyres weren't amazing, as you'd expect when the whole lot cost £70 including the wheels, I'd still say they offer a fair bit better traction than mountain bike tyres - obviously mountain biking people tend to worry about rolling resistance and rotational inertia a good chunk more.
I'd say wheelies require a fair bit of torque, but often less so power, unless you're intentionally keeping them low - so they need low down power then either very little if you're keeping them at the balance point, or a bit less below - the lower down the wheel is and thus harder acceleration, the more power is needed.