Captain: "Computer, Set course two-one-three Mark seven."
Computer: ""Unable to comply, the automatic pilot system is offline."
Automation is all very well and good, but like any system it can fail. Today a plane could take off, fly to it's destination and land all by itself." But would you fly on it without a pilot?
As for having a dedicated department, sure everyone might receive basic training in piloting a starship, but their is a difference between basic and advanced training.
I hear what you're saying, but for me it was a case of not having really watched TNG for years, and getting a sense in the intervening years that tech has moved on here IRL and so feeling the ship should be able to fly itself. I still think it could. As for wanting to get onto a plane that could fly itself? No way!
I don't think this is so much about the a ship being engineered to fly on itself, but the question wether humankind wants to step away from using
technology and instead getting fed by it.