The article mentions that the researchers factored in wealth, to allow for the possibility that the correlation might be more 'have plenty of money to spend on shows and other cultural events = less likely to be depressed'.

But there are plenty of other factors involved.

First, relative isolation: the ability to go to cinemas, theatres, museums presupposes living close enough -or with easy transport access- to a well-developed town with good amenities, rather than an isolated rural setting. That proximity and connectivity would in itself be a major factor in the mental health of a social animal, I should think.

More significantly, people tend not to attend cultural events alone - they go with like-minded friends.

So 'regularly goes to museum/theatre' also means 'has at least one, probably more, people in their social circle that share specific, subjective interests'. And that is a massive factor: a social circle or household with whom one can actually converse fruitfully.

If those living alone and visiting museums, etc. alone also revealed the same benefits, I would feel confident ascribing the positive effect to the cultural stimulus. Otherwise, I would tend to imagine that the social side was also playing a major part.

You could have a control group engaged with the same frequency in non-cultural social activities (e.g. sports or other hobbies). But I would want a proper statistical analysis of the social aspect.

Perhaps the paper in question also covers this, but I don't think it's mentioned in the article.