Sam: I’m addressing you because you are rational.
So, I imagine you are correct that it is not assur to be say hello to a woman.
But, all that means is that it is not part of a hard rule. It is obviously still subject to a question of what we can see that Hashem wants us to do.
It is quite clear that chazal frowned upon the idea of men and women having relationships. Of course though, it is a question of line drawing. I don’t think it is far fetched at all to say that men and women should not have the sort of relationship where they greet one another in the street, and should not greet one another in the street in order to avoid such relationships.
Before reading this thread, I would have thought it is appropriate to greet a woman on shabbos in the street. But I’ve been convinced otherwise now. I don’t like it.