I guess that in some cases that 'impact' could matter, particularly in some very crude mechanism....

But the problem I see here, that during that first 'impact' string can really transfer rather tiny amount of it's energy to arrow, and even if there was some 'shock' caused, then string still has a majority of it's energy, and time to transfer limbs energy in 'straight' direction.

How big would be the difference from bow be - arrow pushed 'instantly' vs few millimeters of string traveling on it's own.

Hard to say.

1. Perhaps the authors weren't as trained in the use of bow as they needed to be? It is generally agreed that learning to use a bow effectively was basically a life-style, something begun at childhood. On the other hand, the crossbow was something that could be trained in a comparatively short amount of time. If your "example" bowman was lacking the background of the former, then we could expect a crossbowman to out perform him.
I would suspect that simply no matter how trained bowman is, the very nature of bow shooting is somehow more erratic than situation where one doesn't need to hold that string.