Barbarians don't tank with AC, they tank with HP and DR. This is a deliberate 5e design choice. If you want an AC tank, there are other classes for it. Losing barbarian benefits if you wear armor is "working as intended". But yeah, I wish it was possible to lock in "thematic style" for all characters regardless of what they're wearing.
They don't lose any Barbarian abilities wearing up to Half-Plate. That's 17 AC compared to 14 or 15 AC unarmored with all Barbarian abilities working just the same.
There seems to be some confusion on how a Barbarian "should" be played, also at the Larian offices when they don't put a low level Barbarian in Medium Armor. Unarmored Defense is a trap that's working very well. Medium Armor is a strictly better choice unless you find some equipment that gives you benefits that rival the lost AC, or if you have 20 Con.
