From my understanding, you're asking if you should have tracking from youtube to your website, or if Google Analytics will show that traffic based on how much traffic your YT is getting.
If that is the case, then I would suggest tracking them individually as you said. The Google Analytics tracking only tracks how many people go to your youtube, not how many people go to youtube and click your website link.
Therefore, Google Analytics will not show traffic going from youtube to your website, only your traffic to youtube.
Hope this helps.