No, it shouldn't. Runner still has to wait for the ball to leave the pitcher's hands.
In an effort to curb pitchers crow hopping, there was a season in the early 2000's where an experiment was tried allowing the runners to leave as soon as the pitcher started the forward motion. This meant the runner would be 3-4 steps towards 2nd by the time the guy jumped, landed, and threw the ball. Basically, anyone on base was almost guaranteed to get to 2nd easily. Not many runners were thrown out!!!! But, that went away pretty fast due to the absurdity of how many runners were able to steal.
