I think our society expends a lot of energy and political capital trying to make equality within a single generation possible. I believe that will never work. Who you were born too will always be a huge influence on where you end up on life. In addition, one of the biggest motivations for parents, is to give their children every advantage that they possibly can.
However, instead of fighting with this parental desire to better our children, we should use it. While it may not be possible to achieve equality in a single generation, it should be possible in 2-3 generations. Many immigrants have demonstrated this. The first generation may only work low-paying menial jobs, but they provide enough of an education for the second generation to get higher paying jobs and enter the middle class. The second generation works and saves and their children attend elite universities. The third generation is part of the upper middle class, and with the right connections and luck, they are their children can enter the top class of society. Thus we do not have to do the Sisyphean task of trying to make equal outcomes for everyone, we just have to work to make sure that each parent has the ability to make life better for their children.