Utah — “Chazal made statements… I don’t think that they made statements about not saying Good Shabbos to the opposite gender”
Incorrect. Chazal definitely did make such statements, as quoted by posters before – ??? ?????? ????? ??? ???. A simple transalation of this is that a man is prohibited from “asking peace” – aka greeting – a woman. (Halevi before explained that it is not necessarily the intention of the words, but the statement is there to debate).
Anyways, “#takingbeingfrumtofar” is a bit harsh – not everything Chazal had to say explicitly; the Shulchan Aruch says “Hisrachek min Hanoshim” — distance yourselves from women. The Gemara countrless times numerates the issurim of having too much contact with the opposite gender. Whether greeting is specifically ossur is probably irrelevant; if it causes Hirhurim, it is Ossur.