“1. He did not grow a beard.
2.The story is true — and not so unbelievable.”
There aren’t enough details for me to assess the likelihood. If his shailah was, “should I grow a beard and peyes,” then, yes, it is very believable. If his shailah was, “XYZ happened do this dish, is it still kosher?” Then it’s a little harder to believe. However, I’m assuming he went to a Chassidishe Rebbe with some kind of generalized, spiritual guidance (non-halachic) question and was taken aback when the rabbi gave an actual halachic shittah; is that what it was?
For what the OP meant by “gadol hador,” I think it can be pretty simply explained as “who is the Reb Moshe of today?”
For what I said about the MO, let me try to lay out an imaginary flow chart. First of all, there is a massive portion of the MO (I believe the majority, but you would know better) who simply never ask shailos ever. Clearly this thread doesn’t apply to them. Of those remaining, a significant portion may ask shailos to their local, MO shul rabbi and be content with his on-the-spot psak without any proofs from major poskim. This thread also doesn’t apply to that group. Finally, you are left with a tiny niche in the MO that actually asks shailos regularly, and expects the psak’s to be well-based in halachic reasoning, and escalated to more experienced poskim if need be.
This niche chooses to self-identify as MO, even though for all intensive purposes they aren’t. We can’t be expected to use this niche as the basis of all generalizations we make about the MO just to spare feelings.