Sorcerers Apprentice
Hero
Garbage archers? A Battlemaster fighter with Sharpshooter and Precision Attack is possibly the best archer build in 5e. High and consistent damage, and can be combined with other tricks, like Crossbow Master."Just the first phb" isnt a meaningful qualifier. That isnt how 4e works. the first phb does not represent the base game. if ypu dislike that, fair, but it is a fact. The first phb isnt more important than the second phb.
The 5e fighter cannot be effecfive in any role than front line damage tank. Archer? They're garbage archers. 4e had much more actual versatility and flexibility.
The only thing 5e fighters are naturally good at is hitting hard and being tough enough to withstand enemy reprisal, ie tanking.
Not as good in 5.5 because of the nerf to Sharpshooter, but I bet it can still work.