In Michael Swan's "Practical English Usage" I read: "We very often use "at" before the name of a building, when we are thinking not of the building itself but of the activity that happens there.
"I first heard her sing at the Usher Hall in Edinburgh."
It's interesting that the example uses both at and in.
The book gives another example and says: "It was warm and comfortable in the club." (A place to spend time). I don't understand its comment. I think it's more helpful to say as it was explained here that "in" is a restricted area.
Yes. The restricted area is the club's interior. Inside the club, it's warm and comfortable.
You could also say it was warm and comfortable at the club.
With in, the sentence suggests that someone is actually experiencing the warmth and comfort. Jerry didn't want to go out into the blizzard. He was warm and comfortable in the club.
As you say below, with at, the sentence is describing the club more generally: Lucia had several reasons for wanting to join the club. Many of her friends were members. And it was warm and comfortable at the club.
I may be wrong of course.
So "at" is used when speaking about a place generally and also when speaking about the purpose being there.
I think you have the right idea.
