Why can't I kill them?

you can, you have to deshield the dent first

Assuming your suggestion is that we should be able to kill them, I agree.

I also think they should be permanently lost until the ship is repaired in a station.

According to a support ticket reply:

Trident turrets currently are indestructible while the owner is piloting the trident, even if shields are down. They blow up fine if the owner isn't currently docked.

Yep, and they regenerate when the owner docks, which is dumb.

A trident without shields turrets are still non-killable, if you dock/undock fast enough to can stop swarms from tracking on target.

This needs to change.

Now if this isn't broken and a exploit, then I don't know what is.

It's both.

I'm surprised I was not aware of this particular broken game dynamic. This certainly needs to be fixed. Overall the trident has several things that need to be adressed:

1: invincible turrets (as OP and Waldo suggested)
2: seeking munitions breaking lock during dock/unock
3: exploiting dock-delay to jump sector regardless of cell capacity (I'm looking at you, TGFT ;] )

Adressing these issues before releasing new capship variations would help save the devs a lot of player-anguish and backtracking in the future.

This suggestion post is gold.

Bump