The in-universe answer re: drones would be that people want to explore. Sure, it’s dangerous, but it’s also exciting, fascinating, and fulfilling. That said, I feel like a responsible captain would make much more extensive use of probes than any of the shows.

Re: data streams, I don’t have a good in-universe explanation. I have a similar question of why they don’t have security cameras in all the hallways and public areas.

Also, using the transporter to go down to a planet always runs the risk of some storm or an orbital threat stranding your party. Why not use the shuttle as SOP? It gives your away team more resources, both for their mission and for an emergency.

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).

No one’s going to watch a realistic exploration sci-fi show about small unmanned ships quietly going about their jobs with no drama.