A useful response to this would be to then explain the structural issues that cause the higher poverty rates rather than assume the middle class person in question is arguing from a position of ignorance. How is someone supposed to respond to "check your ignorance"?
"Check your privilege" is a metaphorical shorthand asking the person you're talking to to listen to your arguments with a critical eye towards how their own identity shapes their pre-conceived notions in the debate. It is typically accompanied by other arguments which attempt to illustrate what the pre-conceived notions are. Very rarely does someone say "check your privilege" in total isolation, unless the thing they're responding to is deemed so obviously outrageous that it doesn't require a sincere response--in the same way that someone might reply to a creationist with "Science does not support creationism without necessarily walking them through the
how of it.
For example, if two people were discussing the fact that "anyone who didn't learn to swim as a kid is basically an idiot who failed at life", someone might interject that there are privileged assumptions embedded in that argument. Learning how to swim requires access to a source of water. In a rural context, you likely have outdoor swimming options. But in an urban-suburban context, access to swimming facilities is dramatically better if you're middle class or better. First, because if you're middle class, you're much more likely to have a pool or know someone who has a pool. Second, because if you're middle class, you're likely to have more secure transportation inside an urban area, and thus better access to public facilities. Third, because if you're middle class, you're likely to be living in an area with safer public facilities in a better state of repair. So, yes, it's true that the best way to respond is to lay out the obvious case for why their statement is ignorant. But imagine that one of the people did not know how to swim, because he grew up in a bad area of town with limited resources, single parent who worked two jobs, etc. The original statement itself was pretty rude and offensive, and it's logical that the person's immediate response would be "Dude, stop being ignorant. Not everyone learns how to swim." And it's true that that response is incomplete and not necessarily productive, but it's hardly worse than the original comment and it's likely to be a part of the broader response and explanation.
If you were being insensitive, being told to check your priviledge tells you nothing about how you're being insensitive and what experiences those oppressed people have that you aren't considering.
In part the problem here is that you've taken the following scenario:
Ignorant perpertrator says something rude to innocent victim, doesn't consider the impact of statement on other people at all
And you've turned it into:
Insensitive victim fails to spend time explaining how perpetrator was ignorant
If we're going to police tone, the burden should be on the original person to constantly be empathetic, kind, and open-minded, not on the person negatively impacted by a rude person to explain why. It's true that the latter is probably a useful approach and might result in positive discourse, but it's hardly a reasonable expectation.
