There is nothing "bureaucratic" about what you're talking about. Rich people like the art they support. It is not just about status. Or else it wouldn't matter what kind of organization they donate to, but that's not true. Many rich people care about status but don't support the arts at all.

It doesn't follow that because not all rich people who care about status support the arts, support of the arts by rich people is not just about status. Not everyone takes every possible action in support of their goals, indeed, for something like status, with so many avenues of pursuit, it's unlikely that anyone does. I'd be willing to bet that more rich people care about status than give to charity as well.

I don't doubt that there are rich people who care strongly about the arts. There are certainly non-rich people who do, and I don't think richness would filter very strongly for people who don't care about the arts. But I think you underestimate the importance of status signalling; being seen to be heavily involved in a cause is a stronger status signal than being seen to merely donate.