I agree. Even with the mistake corrected, I dislike the way the sub-class is executed, precisely because you get so few arcane shots. It's even worse if you miss, because than you've used one of your very limited uses. Personally, I would have given the arcane archer a kind of Eldritch blast type arrow cantrip or something, so they can cast very basic 'magical' arrows every turn.
I have the same problem. I'm ok with AA being a figther archetype, but it feels more like the Seeker from 4e than the AA we've seen in previous editions; they feel more primal than arcane. I feel like the gunsmith archetype for the artificer UA does for a better AA ir refluffed, or even the horizon walker with all the ''arrow spells'' the ranger gets. I would have preferred an archetype that gave us different at-will action type to let the fighter do something different than ''use the Attack action'' every turn, like the gunsmith. Something like this:
lvl 3:
Master Fletcher: Gain prof with carpenter tools
Arcane Quiver: Can craft an arcane quiver with special leather and you can use it to craft +1 arrows (+2 at lvl 18). You craft 20 after a long rest, 10 after a short. Can replace the quiver for 30 gp worth of material.
lvl 7: Phasing Arrow: as an action, can shot an arrow that ignore 1/2 cover and deals +1d6 force (+1d6 every 2 lvl thereafter)
lvl 10: Hail of arrows: as an action, create a 15 ft cones of arrows. Dex-save or 2d6 (3d6 at 15, 4d6 at 18) force dmg.
lvl 15: Piercing shot: as an action, create a 5x30 ft line, Dex-save or 4d6 lightning dmg (6d6 at 18).
lvl 18: Exploding arrow: as an action, create a 30 radius blast, Dex-save or 4d8 fire damage.
This gives you alternate options for a fighter turn, instead of the classic ''attack and maybe pile another effect a few time per shot rest''. You can decide to use your fighter multi-attack if enemies are spread out or use a special shot when the battlefield allows it.