it’s good that you told the player your thoughts!
“It’s in character” defenses are rarely worthwhile. There has been a lot of (some good, some bad) rpg theory about the social contract of playing RPGs. The things you describe doing would be fine in a lot of games/with a lot of groups but would fall afoul of some groups’ expectations. ie that players won’t have their characters take intentionally stupid actions even if those actions are entertaining.
5e is popular in part because its… well its not a blank slate, but its a grab bag of game stuff that can be bent into many shapes. It’s also popular because it’s called D&D. Anyway, because its so group-neutral it has the drawback that the game doesn’t give you any tools to support a particular playstyle.
The ideals, bonds, and flaws are supposed to help with this but I find that they’re somewhat underwhelming in practice. They should serve as flags of PC behavior but they’re just invisible to the other players from what I’ve seen. The ones in the book are also often so distant from what players actually do in a D&D game that they feel extra-useless. If your barbarian had the flaw of “Charges in without warning” or something like that, it would at least have been clear to the table.
I feel that in a good game the people at the table can lay out some ground rules for what they expect from the game, like “we don’t like pvp even if it makes sense in-narrative”