The wonky AI definitely makes the bird seem duuummmm. The healer bird stays by you because it is not engaged with any enemies. Since there is no other AI programmed in, the bird is dumb and does the only thing it knows... Follow you everywhere.

To better keep your Bo alive and out of AoEs, you'll have to micromanage its stances. Switch to healing when you need it, and leave it in DPS the rest of the time. You can also send your Bo away before it dies to prevent the horror of witnessing it. It also allows you to immediately resummon him.

In their defense, chocobos are freakin clutch on their stuns. They do it better than most RL players.