Joseph wrote: “Lo Sikrevu – the Torah prohibition of touching – is referring even to non-affectionate touching, according to the Rambam. The Ramban argues, and holds that non-affectionate touching is not included in the possuk, but is assur nonetheless m’drabbanan.”
It is not poshut bichlal to say that the Rambam was referring to “non-affectionate touching.” The poshut understanding of the Rambam is not like that, though the Beis Yosef did understand the Rambam like you, but the Shach disagrees completely (Yoreh De’ah 157:10). R’ Moshe in his teshuva paskened like the Shach. Also, I don’t think that the Ramban mentioned “non-affectionate touching,” either. Perhaps you got that confused with the Teshuvos HaRashba.
Joseph wrote: “TJ, I don’t believe Reb Yaakov ever said any such thing regarding a heter for embarrassment. Do you have a mekor for that? I think not.”
Actually, I do. Emes L’Ya’akov on the Tur and Shulchan Aruch, p.405 n. 4.
Joseph wrote: “Reb Moshe wrote befeirush in the quoted IM that shaking hands is always assur under all circumstances. Period.”
True that R’ Moshe asured handshaking, though he was lomeid zechus for those who hold that it isn’t derech chiba. I just wrote what R’ Reuven reported about his father’s shita, in a case where there is the additional issue of embarrassment.