On one hand, cool demo, on the other, this is horrifying in more ways than I can begin to describe. You're literally one prompt injection away from someone having unlimited access to all of your everything.
As long as it’s gated and not turned on by default, it’s all good. They could also add a warning/sanity check similar to “allow pasting” in the console.
Of course I still watch it and have my finger on the escape key at all times :)
Not the person you're replying to, but: I just use a separate, dedicated Chrome profile that isn't logged into anything except what I'm working on. Then I keep the persistence, but without commingling in a way that dramatically increases the risk.
edit: upon rereading, I now realize the (different) prompt injection risk you were calling out re: the handoff to yt-dlp. Separate profiles won't save you from that, though there are other approaches.