Streaming video is almost exclusively pull. The only data you're sending up to the server is what you're watching, when you seek, pause, etc.

Useful LLM usage involves pushing a lot of private data into them. There's a pretty big difference sending up some metadata about your viewing of an MKBHD video, and asking an LLM to read a text message talking about your STD test results to decide whether it merits a priority notification. A lot of people will not be comfortable with sending the latter off to The Cloud.