The point here is that it kind of ruins the Romeo-and-Juliet-ness if either family has a reason to hate each other that isn't just the self-perpetuating nature of revenge. It has to be the case that, at any point, both families could just say 'you know, this is stupid actually' and call off the feud, which is of course what they do in the end.
If instead one house is the haves and the other is the have-nots, the enmity between them has a very different character, and thus the relationship of the Romeo and the Juliet will also be very different. You can no longer treat that feud as irrational! And you definitely can't treat it as symmetric. If you have rich Capulets and poor Montagues, you can have the Capulet side of things stay roughly the same, but it's very hard to have the Montagues reject the relationship between their son and a girl who's filthy rich without coming across as just plain stupid.
I blame West Side Story. Which is a good story, don't get me wrong, but it's crucially a transformative adaptation of R&J, and not R&J itself. If your supposed Romeo and Juliet story also comes with a heavy dose of ethnic tension and unevenly-distributed police violence, then it's not a Romeo and Juliet, that's a West Side Story. Same reason you shouldn't call something a Hamlet adaptation just because it's got lions in it.