State universities kind of do have this responsibility. My school is doing more or less this, restructuring some of the math sequences that have a high failure rate, like precalc/calc 1.

Also state universities are where most of the public school teachers and administrators trained, so they probably deserve a little of the blame for the state of public schools.

IMO the only stuff a student "should have learned" is stuff that they can't get onto the course without knowing. Allowing someone onto the course, for which "x" is a prerequisite, who doesn't know "x" is entirely the university's responsibility.

All public schools do not suck but especially urban schools can have a lot of problems and many of the students don't have a great home environment either,