It's all great to want to keep humans in charge of things - but first time you come across an enemy with ships that are piloted - and who weapons are *targeted* - by an computer...you are gonna be outmatched.
When your weapons officer takes many seconds to target weapons - and only after he/she has been ordered by the captain to - against a ship that has targeted and fired in the first *nanoseconds* it encountered you - you are gonna be toast. Your ship is gonna be a vapor cloud before your commanding officer can even say "fire." Hell, you are gonna be toast before your human crew has time to *think* "We need to fire weapons..."
And by then - assuming you *weren't* completely taken out - buy the time you target and fire back - the AI-piloted ship is gonna have anticipated and moved out of the way. Or seen the weapons coming - and moved before a mere human pilot's brain could even register "we need to move". Same for trying to get out of the way yourself when ultra-fast AI-targeted weapons have already been launched against you. And odds are they are smart weapons - and can follow you and adjust to any move a human (or other biological-based) pilot can make faster than the human pilot could make them.
Computers are just faster. Much faster. (And Trek's clumsy 2-D flat keyboard-based interfaces, where you have to enter complex commands an moves by *typing* in numbers and letters...just seem horribly antiquated an inefficient compared to a machine than can just think it an the ship moves...)
If you want to keep a human in the loop - fine. But I would make the human crew cyberneticly enhanced (and when you consider that any future human starship crew is likely to be post-human, it seems likely that they *would* be) - and I would have a augmented human-linked to an AI pilot the ship and fire the weapons. That way you get the best of both worlds.
At the very least - I'd replace the clumsy keyboard interfaces with thought-controls. But even those will be slower than an AI-controlled vessel - or even a vessel where a cyberneticly-augmented human is linked to an AI. (I am assuming some sort of tech magic where the cybernetic enhancements would allow the human to think as fast as the computer...though that would probably mean that some of the human's thought processes are ran on or in the computer...)