I don’t think the but n ben the Broons visit is a Bothy. Iirc they owned it (or at least rented it) and it was pretty well kitted out. Bothies tend to be pretty Spartan and are just freely available, not owned or for hire
Random aside - the But n ben is also the name of a nice little restaurant a little just outside of Arbroath.
Quite a few places I heard called bothie were privately owned by curious share communities. Up by the headwaters of the iron bru springs around moffat (I know... not really but it was amazingly ferrous water) there were a few shacks we called that way. And down cockburnspath.
Sure. Up in the hills, for serious climbers and walkers? Different story. But bothies included Shephard's huts, and working mens fishing camps and all kinds of things.
What do you think "the butt and ben" looked like before maw broon got some carpets off the rag and bone man and paw broon got that auld sofa?
If you want to be pedantic, yes the working class lowland "hutters" were different. I guess that's what I'm talking about. Colloquially people call these places bothies. It it's not part of the MBA hill stuff, for sure.
I'm pretty sure they are all privately owned by the landowner (usually a sporting estate, although some are in areas owned by conservation groups e.g. Ryvoan) - even if they are maintained by the Mountain Bothies Association.
NB The default in Scots law is for any building to belong to the landowner even if someone else builds it so even if someone did build a hut somewhere it would still end up being owned by whoever owns the land.
Random aside - the But n ben is also the name of a nice little restaurant a little just outside of Arbroath.