I was responding mostly to the "popular media" claim, but I take your point.
The question of a "cultural push" is interesting to me for two reasons:
1. It suggests that there's been a concerted effort among liberals to reshape culture as a whole, and,
2. It makes broadly agreeable political positions more partisan than they need to be.
I don't know of a way to prove or disprove (1), besides saying that I'm not on the mailing list. I think progressive views tend to be more fragmented and diverse than conservative onlookers realize, and that this contributes to a feeling that progressives are advancing a monolithic program to change America's culture.
On the other hand, I think (2) is pretty worrying. Are gay rights part of the cultural push? What about access to (reasonably priced) medicine, or environmental consciousness?
I think that, apart from the true-believing fundamentalists and hardliners, most people can recognize that many progressive positions are at their core very agreeable (civil liberties for all, no unavoidable deaths, modernize our economy both for our own health and for future generations). The problem arises when those positions become associated with progressives qua some conservative boogeyman, not qua their benefit to society.