"Club" can mean both a private members club with various facilities, including a bar, as well as a place where people party and dance all night to loud music.
Why try and look for a hidden meaning in something that is obvious? That sounds exhausting.
It's weird enough that Musk claimed he was sad-drinking in a club, but now we are discussing the idea of him sad-drinking in a country club, a place for old, rich, white men who go there to play golf - where he was then approached by a young model?
There are many kinds of club, and there are many reasons to have a drink by yourself, most of which are not about anybody’s sorrows. Maybe you like the music, for example, but aren’t in the mood for company.
But if we want to assume Musk was drowning his sorrows, and that it’s an EDM dance club — this seems to be your implication, maybe it’s true — then I still don’t find it weird. Darkness, distraction, the desire to be among people but also alone, avoiding your entourage… plenty of good reasons.