Does university have any cultural mission?
Almost all universities have a larger mission than simply provision of educational courses.  For example, consider the mission statements of two very different publicly funded American institutions:


The University of Texas System, in addition to education, aims to develop ethics and morals, appreciation of arts and culture, and prepare "educated, productive citizens who can meet the rigorous challenges of an increasingly diverse society and an ever-changing global community"
Bunker Hill Community College is much closer to the "pure skills" model, but still also aims to provide community leadership and to ensure that "no eligible student shall be deprived of [education] because of an inability to pay tuition and fees."


In both cases, the university is explicitly aiming to promote social and cultural change towards a society in which most people would generally prefer to live (even though the particular goals of each are different).

Now, mission statements are one thing, reality is another.  Every university's leadership decides how it wants to interpret and/or modify its mission statement. Universities also generally have some sort of board overseeing it, which will attempt to guide and shape the university's actions.  This is where, especially with public universities, there is often a great deal of fighting about the university mission and policies around it.  For example, conservatives who are uncomfortable with the cultural broadening provided by universities often argue that a university should "just stick to teaching skills."  In general, however, the faculty and leadership of most universities tend to promote some sort of cultural mission, even if some are very different than most.