New Zealand has about 1000 of these, but we call them huts.
In case anyone is wondering, it's pronounced 'baw-thees' not 'bo-thees'
I stayed in a fair number of them up in the Scottish Highlands last spring
A lot of cottagers in Canada I've met keep the key by the door in the winter. Reason explained to me is snow mobilers and hunters will look for places to stay, either for convenience or emergency, and if you don't leave the key they'll likely break in anyway and if they have to break in they'll probably treat your place like shit and destroy it but if the key is there they'll probably take care of it as a thank you.
I've stayed in one of those before. They're quite nice, but holy tits the rats were not afraid of people.
I’m all proud I knew this.
(I had looked it up after hearing the line “...duck under the eaves of the bothy...” in “Yon Two Crows” by Mark Knopfler).
We used to use the near by one to get messed up drunk in :-)
Do not attempt this in the US if you don't know the area and aren't absolutely certain it's a publicly accessible abode. That's how you get shot in some places. (just a warning, I'm nit going to shoot you, but backcountry folk generally want their privacy if they live out isolated like that, and especially in the rural South, self defense shootings wouldn't be unusual.)
Still doesn't explain those random sets of stairs you find.