Hmmm, Yaakov Avinu and Moshe Rabbeinu met their wives on their own, as did Dovid meet Batsheva. Secondly, as pointed out, dating that could potentially lead to marriage is not what Reb Moshe was pointing out. He was writing about an elementary school or HS kid. Even in coed high schools, many have met their spouses, and gotten married later, so there is always a possibility of eventual marriage.
Does anybody seriously believe here that meeting one’s wife on his own is an issur? (The gemara which says mekadesh blo shidduchin refers to meeting somebody and eloping on the spot.) If one meets and then plans a wedding with full parental and family involvement, that is not a problem. I have heard of cases of Yeshiva boys and girls who gave one another a ride to a simcha and ended up getting engaged. It is so ludicrous to think one cannot meet somebody by chance anywhere. That is how Hashem runs the world. Many BT families have no connections in the shidduch world, and their kids have to meet somebody at a social event or similar. A number of years back there was a frum singles social group (forgot name) where there were Rebbbetzins floating around to introduce if people were interested in somebody there.