Coercion, fear and force can all invalidate a marriage, because it has to be undertaken freely.
I don't get the sense that the priest "refused" to marry them under any circumstances, though, especially if you look at what the teacher says about it.
That's just unwarranted hyperbole by a clearly biased writer.
You're right. It is the norm for a couple who has conceived out of wedlock to be made to wait until some period after the child is born before being admitted to the sacrament of marriage.