TwoSix
Everyone's literal second-favorite poster
Reckless Attack only works on melee weapon attacks that use Strength. So no advantage for an archer barbarian. Considering the only benefit to rage that works for a Dex-based barbarian is the half damage, and an archer should be taking far less hits in general, there's really not a lot of positive synergy there.Rage still works perfectly. You get all the benefits of rage except the weapon damage bonus. The extra damge is the only thing you miss by going finesse melee, and that damage is significant, but it is hardly an entire chunk of the class design. With extra attack, advantage and brutal critical you are still doing reliable, substantial, damage with a finesse weapon. Other things like damage resistance, extra movement, advantage on dexterity saves, ferrell instinct - they all still work the same and unarmored defense (another key barbarian class feature) works better than it does with a high strength.
A sword and board plate fighter with dueling at level 8 is doing 17 DPR on average vs AC 15 foe. A finesse Barbarian built as above with a shield has the same AC and is doing 18DPR with advantage. Additionally he has advantage on dex saves, a lot more mobility, more hps, more skill proficiencies and can rage for half damage and comparable strength-athletics checks. A basic 1d8 Rogue is doing 17DPR if he is getting sneak every turn but is not getting advantage. Drive AC above 15 and that swings even more in the Barbarian's favor.
If you go archer you miss out on a bit more damage, and you can't keep up with a fighter in damage, because you don't get the fighting style. You will be about 2 DPR behind at 8th level (more if the fighter took sharpshooter) . But you still have all those other perks and not only does rage works fine, in addition it is easier to stay in rage because you don't have to worry about not having an enemy within movement to get in an attack in. As long as there is an enemy somewhere on the battlefield you can shoot at him and stay in rage.
There are ways to make Dex based barbarian work, but it's generally better as a dip to get Unarmored Defense and Rage 2/day, and then focus on a better Dex class (like Rogue or Fighter).