Yes Swing finish makes a big difference. Ichiro had the fastest home to first times of the league all the way to his late 30s. He was fast and being a lefty helped but once he was past age 35 there were certainly faster guys.
But he basically started running at contact while he was still following through while griffey would follow through high over his head and then down to the catchers feet with one hand while also leaning back with his upper body
That high and long follow through and especially the lean back is not conductive of starting fast (sprinters lean forward at the start).
Griffey probably lost about a third of a second to ichiro before making the first step.
Of course the fast start of the box was part of ichiros skill set as it meant defense has to play in so there are more holes to hit through. However ichiro also only hit like 7-10 HR a year so there was not much to lose. Griffey on the other hand hit 45 bombs per year and used the lean back and high finish the lift the ball more. It might have cost him some hits as defenses could play him deeper but who cares if you hit 45+ hr?
For some types fast starts out of the box can help but for most it is not worth sacrificing swing quality by cutting off the swing.