I've always said that Starfleet is, first and foremost, a jobs program.
It gives purpose to people who can't find their own, in a time where your needs are provided-for by default, and seeking personal fulfillment is the purpose for most people's lives.
Drones would cut out the human driving a shuttle over to inspect an anomaly or object themselves, robbing them of a sense of accomplishment and achievement. Starfleet is about that stuff, so that's a no-go unless nobody wants to do it and it needs to be done anyway. We see that a lot, too. They do have probes and sensor stations and stuff, after all, usually in really boring and unfulfilling locations.
They have excessive, ridiculous redundancy. They have people doing jobs the ship computers could (and often, in times of need, DOES) perform very well on its own. There are several recorded instances of entire starships being successfully maintained for extended periods of time by a single individual (who does go insane due to isolation every time, because plot).
starkraving666