Chocobo AI will only attack the target which you attacked first atm. It will stay on the target until it is dead or you flee from it.
If you attack two targets, the moment you kill the first target, the chocobo ai will need another input from you to confirm it shall be attacked. This is why it's running back to you, being in an aggro state while doing nothing besides healing. If you attack the second mob again, it will attack again.

I know it doesn't make much sense, but the idea is probably not to stay in battles which you don't want to fight. It could be that you only tagged a mob that is fighting another player, your chocobo would stick to it, making running away annoying. Another thing is cc, chocobo ai actually tracks if you use cc like sleep and disengages then. Just attacking one target at a time works better for the AI to not break your cc unintentionally.

The "under attack" animation is solely because they are healer. Depending on their level though there might not be many actions they can use yet. Resulting in looking funny while doing nothing.