Who among the parents fighting so hard to get their kids into a good school is going to volunteer to have their kid give up the slot in the upper middle class? People are willing to accept a certain amount of slippage, but only as long as it comes with added job security (government) or special fulfillment (the ministry, the arts)--and even in the latter cases, Mom and Dad will often be strenuously arguing against following your calling. But how many doctors and lawyers would simply glumly accept it if you told them that sorry, junior's going to be an intermittently employed long-haul trucker, and your darling daughter is going to work the supermarket checkout, because all the more lucrative and interesting slots went to smarter and more talented people?
The lack of a safety net makes falling in social status a serious problem. When basic medical care is linked to being a productive member of the middle class, people are willing to make enormous sacrifices to protect themselves from falling in social class. If we mitigate the problems and torments of extreme poverty then maybe it won't be seen as terrible to have people shift around in social class.
Now it is true that my focus is on absolute levels of deprivation. But if the you can be poor with dignity and have basic needs met then maybe that will make us a little more willing to address inequality in general.