When it comes to being countercultural, the Catholic Church's stance on contraception takes the cake: no artificial birth control, no intentionally sterile sex, nothing. More importantly, such counterculture isn't opposed simply to "secular society;" it's also countercultural for many with deeply held religious beliefs. Yet on this question, the Church makes no bones about its teaching: contraception is a moral evil, and no one should be using it.
Perhaps as a shock to some, the Church's staunch discipline, here, hasn't dissuaded many evangelicals from adopting similar opinions (here are a couple examples)—sometimes with greater zeal than that of non-contracepting Catholics. Has the Church caused this (re)discovery of an holistic sexual ethics? Probably not. But its witness hasn't hurt, and has even kept the fire stoked over the past few (very chilly) decades.
It's a similar case with cohabitation. Despite a 1,500% surge in shack-ups over the past fifty years, the Church's injunction against living together before marriage (and its inflexible notion of scandal and near occasions of sin) isn't going away. Yet here, too, we find non-Catholic and even secular opinion growing in opposition. I've written on this before, employing the thought of secular sociologist Zygmunt Bauman. This article from the The New York Times is also fantastic. Once again, has the Church caused these awakenings? No. But it might very well have made them possible.
All of this, coupled with—according to a former director of the World Bank—lifting "more people out of poverty than any other civic institution in history," makes "illiberal Catholicism" a pretty easy sell. Or at least it should. Because neither the Enlightenment nor the Inquisition is anywhere in sight. And committing firmly to wisdom rather than convenience tends, in the end, to make a lot more sense.