My best understanding is that there are too many Pizza shops for a single Menahel to cover his own students.
and,
If the menahelim of the various yeshivos will be at these pizza stores, then they can watch for improper mingling. Therefore, there is no reason why a bocher can’t get a slice after a few hours Motzei Shabbod (or Shabbos afternoon) learning in the BM.
I think these two posts answer each other.
The point is not hashgochah temidis. The point is mirsus.