Socialization is best taught within the family structure, and to a lesser extent the community.
Schools should adhere to and reinforce societal lessons, but should not be the main source for them.
Dream away, but there should not be a need for schools to focus on social well-being and if there is it is because the people who long for this have worked diligently to destroy the fundamental societal structure that for thousands of years has served this purpose.
Schools should educate students on the facts and general concepts they need to know to thrive in society and appreciate the wonder of the world in which they live. This does not include how to care for one another, how to behave in public etc.
If a child is berift of familial and communal sources of education on social interactions, it is because progressives have, intentionally or otherwise, done their best to undermine social institutions.
By arguing that schools should, primarily, concern themselves with the social well-being of their students, you are arguing for the dystopic future you fear.
Yes, there are children who do not have the benefit of familial and communal institutions, but putting aside (for now) the fact that progressive ideology has exacerbated this problem, schoold are not the place to fill in the gap; particularly when not all (or even most) of the students suffer from this deficit.
I would be happy to learn that progressives are now arguing that socially deprived children must be sent to "special" schools where they might receive the societal education their crack-head mothers, absent fathers, and Project communities have not provided, but this will never happen, as the longing to be politically correct will always triumph.
If being civil is, indeed, no longer a natural part of the social order (and a good argument can be made that this is the case) it is because of progressive ideology that raises the State above all other human institutions.