As for things like “singles’ groups”. Many churches of any size will have a young adults group (18-30, more or less) and these often attract the 30-plus singles. Partly because one is more likely to “find someone”, especially in a large church, but there are other reasons.
What also happens, I have observed, is that there is a lot of informal mentoring; with the older single men helping the younger ones (I benefitted a lot from the help of one man who didn’t marry until past 40), and the older single women doing likewise for the younger women. The church I belong to does offer a formal programme to encourage mentoring (discipling) of this sort.