Guild Wars 2

Would it be completely overpowered if there was no increased cooldown when a pet died? What would it hurt?

It encourages pet management skill. Attentiveness and timing.

How in the world do you tell your pet to stand out of the aoe circle while you’re following the meta of melee stack cheese all bosses?

Pay attention to it’s hp and swap whenever low or it’s dps rotation done. Plus, pick proper pets for the encounter, aka not too squishy one and can dish out dps. Drake probably is the most balance between tanky and dps. Ferals bit squishier, you’ll have to swap quicker and more often.
Mai Trin, Thief Group Stealth and Ranger 2xBears strategy, utilize this and show how good you are at managing pets. Same applies to playing your ranger in general. You need to manage your pets.

There is really no good reason for dead pet swap to be 3x longer than normal. No other class is penalized for not micromanaging their mechanic perfectly, nor can their mechanic not function for periods of time due to being dead.