In the VN he did, yes, but the VN did not show an image of Archer's arrow being barely inches away from Berzerker's face, so could have simply started the block while the arrow was a hundred meters away.
I'm not denying Berserker is fast, but in the absence of hard numbers I can only apply common sense. For an analogy, mundane humans do not outrun mundane bullets, nor are they fast enough to block them. Translated to servants, it's somewhat safe to assume that servants are not fast enough to outrun servant bullets, though blocking is perhaps possible due to instincts and magic mumbo-jumbo.
You may not like the analogy, but think of it this way. The Holy Grail War is a game, and as a game it needs to be balanced. Berserker excels at everythin physical while Archer is mostly a weakling. The only think Archer has going for him is his arrows, and if his arrows aren't faster than servants then his class becomes downright puny and useless. He can shoot all he want's but he'll always be dodged. So for the Archer class to remain significant during the game, the most likely abilitie for him to posses is for his arrows to be fast enough to take servants by surprise.
Gilgamesh is not a good example for an Archer. We don't know if Gil was being serious or if he was toying with the dog in that scene.