The point is not hashgochah temidis. The point is mirsus.
Like!! However, then the point is mingling, not stopping off for a pickup or going with a few other bochrim. (from the guy’s side, as I am a guy).
That being said, I don’t think that the issue is specifically relationships. There is also the smoking (tobacco or other items), boozing and plain not-menchlechkeit (stuff between men) that goes on at a motzei shabbos hangout. The menahelim would (IMHO) be doing the community a favor by policing the area for untoward activity. And yes, it is their job, as teachers of children, their job does not stop at the yeshiva gates.