Figure it's worth mentioning a bit of a summary here...
1) YouTube login is Google's unified login. Log in to Google, it logs you into everything (Gmail, YouTube, Google, etc).
2) Google broke Tesla's login, not Tesla. Google has some arbitrary block against any kind of embedded browser. Maybe to protect against bots attempting thousands of automated logins? Who knows, but it's Google's issue, not Tesla. Tesla would have to break Google's rules to "fix" this.
3) YouTube app in the car is merely a full-screen browser window - so it preserves login cookies and everything between the "browser" app and the YouTube/Hulu/Netflix browser "apps" in full screen.
d) Many sites have a Google login option (including TMC). Logging in
using Google seems to use a slightly different login mechanism that Google hasn't blocked - so it works that way to get logged into Google's universal login.
5) Log in with the Browser, to any site that has a Google login button - even if you've never used Google login on that site before - and it'll get you logged into everything: Gmail, Google, and of course YouTube.
8) Once logged in, you'll probably never need to log in again. It stays, basically permanently.
B) Plex is only one of countless websites with a "Log in with Google" function. You don't have to use Plex at all for anything, but it's nice.
0) You might also like
A Better Theater (open that site in the car browser; it'll launch theater) to let you use nearly any video site in full-screen theater mode. Browser video codecs are only enabled in Theater mode for regulatory reasons, so you can do things in theater mode that you can't do in browser mode.
9) You might not have noticed there isn't a 4, 6, or 7 above, because the world needs more clearly labeled humor for everyone to enjoy.
Hopefully this helps people jumping to the end of the thread looking for a solution
tl;dr: use the Browser to go to any site (like this one) and use the "log in with Google" button to get logged-in to Google. Then, you're automatically logged in to YouTube as well. But don't blame Tesla, because it's Google's fault.