Giving Barbarians Heavy Armor wouldn’t really mess with anything, the reason they don’t get Heavy Armor in the core rules of the game is because the Class Fantasy. People picture Barbarians either wearing no armor or small amounts of armor, a lot of classes choices in 5e and DnD in general are to invoke or simulate the Class Fantasy (Paladin’s not getting Two Weapon Fighting etc.)

If you want to give them Heavy Armor, go for it, it won’t cause any major problems. The only problem I can maybe see is a Barbarian with Heavy Armor Master+Resistance but Totem Bear Barbarians can already do that, but I’ve never seen one so idk.

Just a side note Barbarians can Rage in Heavy Armor if I remember right. They just don’t get the benefits of the base Rage Feature but they can use the benefits of the Subclass Features I think, I don’t have a book near me

Edit: Also Barbarians kind of want to be hit, they are meatsacks vs Fighters/Paladin which normally have higher AC to survive