I'm just not seeing it. I mean, it could be a fun PC to play... but I just don't see it being that good of an archer.As a Barbarian archer, you will have your Dex and Con pumped, with Unarmored defense that means you can have a relatively high AC without wearing armor no disadvantage on stealth rolls, perfect for you to sneak into the trees or around the enemy to set up an ambush. You get the biggest hit die of any class so you can still stand up to a little punishment and with rage, despite a somewhat neglected Str stat (don't dump it), you can still do decent melee damage in tight combat. If you need an AC bump you can use a shield without losing your unarmored defense.
By taking the eagle powers, the first one allows you to impose disadvantage on attacks of opportunity from movement, so you can escape melee combat with a lower likelihood of being hit, second improvement your eyesight becomes insanely powerful including no disadvantage at dusk and seeing fine detail out to 1 mile, at 14th level you can fly 45 - 90 feet on your turn, to give yourself high ground so you can only be targeted by other ranged attacks.
Added bonus: Enter a rage to take half damage, but by being a barbarian you can melee with the best of them, without compromising being an amazing archer of stealthy awesomeness. Also, bonus points, unique build.
Studded leather is going to give just as good an AC as Unarmored defense, and also no disad for stealth. And Barbs don't have stealth as a skill anyway. (Could via background, or human....)
Grab a rapier and do as much melee damage as a raging barb with mediocre Str.
1 mile eyesight is kinda nifty, but doesn't really do much for archery, you can only shoot about 1/10 that far. Disads on AoO is neat, but you shouldn't be in melee that often. Or take two rogue levels and don't worry about it at all. (Plus pick up SA die and expertise)
You don't get archery fighting style.
You don't get anyway to add to the damage, like sneak attack, or action surge, or hunters mark, or collossus slayer.....
With larger hit die and rage damage resistence... you will be able to tank better. But not sure how much that matters for an archer.