it seems like very few people I hear using the term that way realize that a madrassa is often explicitly secular.

That's not totally accurate. While in some dialects of Arabic 'madrassa' simply means 'school' in both secular and religious settings, in other dialects the word refers only to where religious Islamic teaching takes place, and not secular schools.

Really? I doubt most Anglophones, even educated ones, know that

Well, I did, and my impression is that Steven Pinker is in general substantially more educated than I am, especially where issues of culture are concerned. So I would assign a high prior to him knowing any piece of cultural knowledge that I would. (Although I probably know substantially more Arabic than a random English speaker).

At the same time, there's another relevant direction here: Words can have different meanings than they do in their native tongues, and it isn't unreasonable to use in English the word madrassa just to mean the Islamic universities even though in Arabic the word just means university. In that context, the intended meaning of Pinker's statement is clear. In that context, a possibly more relevant worry is the use of an Islamic example as the go-to rather than say a yeshiva where the point would work even better since both Hebrew and Yiddish have common, distinct words for universities v. schools devoted to religious study. But in that context, Pinker's point is probably balanced by general pithiness given how much most Americans know (I suspect a lot more have heard the word madrassa than have heard the word yeshiva for example).