Could you operate a snowcat privately on public lands? Yes. You would have to register similar to a OHV and follow your public lands rules and regulations.
One problem you would find is that you need roads wide enough to fit a snowcat. Lots of public winter trails (in my experience) go through woods, limiting it to people traffic and snowmobile. Logging roads and two tracks would be the limit.
Keep in mind, the second you travel with your snowcat and plow a logging road, sledheads and skimo bros will follow.
Another problem is location. You would want to live right next to the public trail that your intending to travel on.
The normal way to transport a snowcat on a semi-truck. I imagine Op is not CDL class A certified nor wants to spend the first 3 hours of the day getting to trailhead in a semi-truck.
When it comes to buying a snowcat, its buyers choice.
Older, prone to break, easier to maintain snowcat or newer, lower hours, harder to maintain electronic parts, require computer to diagnosis/certified mechanic to maintain warranty (which is critical if you intend to resell snowcat).
I drove Prinoth and Bombdiner snowcats when working at ski resorts. Great machines but because of heavy use we would put them through, they do break more often then you think.
But maybe if you have the right location, and endless stream of money go here for used snowcats. https://www.skiresortequipment.com/snowcats
Hit me up if you need an experienced snowcat driver. I'd teach you how to drive them.