I think that's a simple semantic issue of who's supposed to count as a conservative. The problem is that the word's been redefined to refer to the classic liberals or those we'd call today "moderates."

So-called conservatives have been dishonest in hiding their prejudices under the liberal banner to make their policies seem compatible with modernism. Or some such conservatives are just loath to use the proper label for their genuinely moderate views, "liberal," because it's been demonized by more aggressive "conservatives." Genuine institutionalists who defend specifically modern/liberal/humanistic institutions or traditions aren't substantively conservative because their values are liberal/humanistic. Substantive conservatives are what I'm calling "medievalists" since the content of their values or policies would take us to a premodern way of life.

This way of distinguishing between liberals and conservatives, cutting through the dishonest redefining of terms, and making the acceptance or rejection of secular humanism (and its implications) the key issue is the way to go, in my view.