I wasn't thinking of the deflection off the fingers as much as the frequency of the oscillations changing with the change in weight of the string. That would affect the point at which the back of the arrow left the string and its apparent stiffness or match with the bow. It's a plucked string and would produce a different "note" if it was on a musical instrument. Separation would happen at a different point on the phase due to the change in speed forward and change in phase side to side.
The frequency of the string does change, but the range over which it changes isn't huge though. I normally get about 108 Hz with a 110 grain string and otherwise similar conditions to the first post. If I switch it for a 70 grain string at the same brace height (therefore same length that's able to vibrate, and same tension in the string) that measured frequency increases to 149 Hz, which is within a few Hz of what I calculate using the equation for the frequency of a standing wave on a string. If I were to increase the mass by 33 grains to replicate the heavier string then I'd expect to decrease the frequency to 96 Hz. A string at 96 Hz takes 10.4ms for one cycle, while one at 108 takes 9.4ms.
That's for plucking the string though, which is the same frequency that the string vibrates at after the arrow has left.
However when the arrow is being accelerated, the string doesn't get to choose how quickly it goes from side to side. In the videos of people like Park Sung Hyun, the time for the string to go from far left (right after leaving the fingers) to far right (right before the arrow leaves) is about 10ms, or the time that it would take for the string to vibrate left to right and back all the way to where it started if that pesky arrow wasn't involved and if the tension on the string (which determines the frequency, and which is at its lowest at full draw) wasn't changing.
The change in mass of the string does have the effect of slowing everything down slightly which is where the effect on the tune comes from. That in turn changes where the arrow is in its bending cycle when it leaves the string very slightly - but it's not because of the change in the string frequency (which doesn't really exist as such during the time that the arrow is being accelerated because the tension and effective length of the string are also changing significantly which both affect the string's natural frequency).