It would be very uncharitable to extrapolate that to "the people joking about eggs don't care about working-class people". The people joking about it largely are affected by grocery store prices. That is, of course, not the point. The "culture war" has a lot of stupid things about it, but also real, unavoidable things that would be unrealistic to expect even level-headed people to ignore.
Edit: Perhaps a more straightforward example: I am personally affected by grocery prices. Some of my closest friends and family are impacted severely by grocery prices. If a wealthy person says to me, "egg prices don't matter", I'm going to say, "Fuck off, they matter." If another person says to me, "Vote for <insert the scariest politician you can think of>, he says he'll lower egg prices," I'm going to say, "because of the egg prices? are you crazy?", even though I think egg prices matter. Relativity matters to actual humans. Some percentage of people in poverty who are desperate enough will feel forced to take that trade-off, especially if they haven't seen the hatefulness (or if they share it, but that's moot), but that doesn't mean the people making jokes about the egg price promise don't care about those people. That's not how humans work.
I put it to you that the conservative opposition and bigots are very uncharitable at the outset, in interpretation of messaging from the other side. In this case they didn't have to do any work, it was literally said for them.
Don't forget that working people and moderates are the ones you need to convince. If people didn't change their votes, election outcomes would always be the same. Educated people tend to be more committed to a worldview (and overconfident), whether left or right. They are listening when you mock egg prices.