MW13 your mekoros can be interpreted another way from the way you understand it.
(you)
Sif katan vav: “Ain shoelin bi’ shalom eisha klal”, which, simply read, means “do not greet a woman at all”.
(moi)
Respectfully, it ain’t necessarily so. It might likewise mean when simply read, ” One does not inquire after the well-being of a(married) woman at all.” The word “Shoelin” means to ask about something. The mechaber could be more literally saying that a man should either not enagage in personal “hi, how are you?” conversation with a woman, or even to ask her HUSBAND about his wife. These are two things that can lead to personal interaction under some circumstances, though most normal people are fully capable of a quick hi how are you without having sinful thoughts.
(you)
“Hevei makdim shalom li’kol adam” could easily be translated as “greet every man”, not “every person”. (The Mishna is obviously addressed only to men; however, one would imagine that the same would apply to a woman greeting a woman.)
(my reply)
Come on! L’kol adam clearly means every PERSON. “Adam” is a general term for mankind. So what if the Mishna (according to you) is addressed only to men (and I don’t know that I accept that to be 100% true). Aren’t women also required to follow all the laws of the Mishnah and Gemarah? Does the Mishna NOT discuss Hilchos Mikvah,Challah, and Neiros? Are women exempt from shmiras Shabbos and kashrus? Are these things not discussed in the Mishna?
As for women greeting other WOMEN – it clearly goes without saying that that this is proper and good, that’s not a chidush. The chidush is in including ALL people. Do you have any idea how a simple good morning or good shabbos or whatever, can lift the spirits of someone who might be ina bad mood? And how it reinforces the good feelings of someone who is in a GOOD mood?
I think that the interpretation of some of these ideas is just a rationalization for some people to exhibit their very poor middos.
When I see someone in the street and say Good Shabbos (and it could be that it is even a woman), and on VERY rare occasions there is not only no response, but they make a point of ignoring me, my thought is “well YOU have yourself a NICE day.”