Earlier quoted context omitted.
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…
Even without the bash escape risk (which can be mitigated with the various ways of only allowing yt-dlp to be executed), YT Music is a paid service gated behind a Google account, with associated payment method. Even just stealing the auth cookie is pretty serious in terms of damage it could do.
But if I understood the original commenter's use case, they're just searching YT Music to get the URL to a given song. This appears[0] to work fine without being logged in. So you could parameterize or wrap the call to yt-dlp and only have your cookie jar usable there.