Plane WiFi is a modern technological marvel and you're lucky to be able to have it at all.
Not so long ago sat phones were the domain of the super rich, because they were paying several dollars per minute. Then it was down to 10 dollars for two hours of multiplexed satellite access. And now apparently it's down to where advertising will work. That's amazing.
Any plane internet I've used has been spotty and terribly slow, but then again, I haven't bought it in years because of previous experiences. I can stand to be without it for 2.5 hours.
Was this 4k video from an arbitrary source, like a random YouTube video? Or from United's website?
I haven't flown outside is Southwest in a while, but they have a bunch of licensed video content that is hosted locally on the plane. And therefore cheap.
No, it's mostly ViaSat (like in OP's picture) now. Gogo never upgraded their infra to handle more traffic and kinda fell out of relevance. Planes with ViaSat will have large oval satellite domes on top that talk up to space.
It's illegal to use your cell phone's cell modem on a plane, because of an FCC rule, not an FAA rule. The cells in the cell network are designed for traffic on the ground. At cruising altitude, your modem can see way too many cells at once.