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.