Bowspammers are a part of the game. They have a great impact on lucky kills and/or wins. They are annoying, although I do it sometimes too. But that doesn't mean you have to remove bows and/or arrows for the starter chests. It's quite simple to avoid bowspammers. Move in a fast, zigzag way towards your opponent. Bowspamming arrows usually don't come further than two blocks, and the max distance that someone can hit (without client modification) is five blocks, but even starters can hit about three blocks away. Also, bows effect 1.9 PvP as well. Since spam-clicking doesn't deal the same amount of damage, it's kinda annoying with bowspammers and the delay between hitting. You can simply avoid that by playing in 1.8 or a version that still supports 1.8 PvP. As simple as that.
Kind regards,
HolyAlpaca
EDIT: Fixed some grammar errors.