“Rav Moshe says in three, separate, strongly worded psakim that shaking hands with women in business is clearly assur.”
Joseph- that is false. Rav Moshe is very ambiguous and seems to go back and forth in his Teshuvos. Rav Dovid Feinstein has said on many occasions that Rav Moshe held handshaking was muttar but didn’t write in a teshuva because he didnt want to be attacked for another Kula. (he was already attacked for many others).