I think you're probably right due to the target audience of the song and who originally seeded its success, e.g. Matt Walsh. Although I suppose you can just argue that Richmond just sounds good put next to rich men? Though if they weren't comfortable with it being associated with the confederacy though they just wouldn't have used it.

If I was to disregard the confederacy association I would say it's an OK name because if you're referring to politicians in DC they're a group that's worthy of being called out. It would still be stupid though because of course many of their favourite politicians are, you know, politicians and also very rich (like Donald Trump but also many more who were born into great wealth and continued to be wealthy throughout their lives.)