Lilmod, Rav Schach was speaking after the fact. We cannot know beforehand how much hishtadlut Hashem wants us to do as a condition for getting something. As there is a difference of opinion among the poskim (google “Shaking hands with the opposite gender? – Mi Yodeya”) she might want to ask her LOR – and he might want to ask her a few questions first.
BTW,shaking with the left hand is considered offensive as handshaking originated as a way to show that one was not concealing a knife (a 2,500 year old depiction of two soldiers shaking hands can be found on part of a stele on display in the Pergamon Museum, Berlin).Perhaps “sinister” comes from the Latin word for left-handed because left-handed people naturally wanted to shake with their left hands.