I tried navigating to youtube-nocookie.com and got an http cert error, so that doesn't seem like an option.

The support page you linked to talks about "Privacy Enhanced Mode". The language there does not sound like it really protects privacy.

> The Privacy Enhanced Mode of the YouTube embedded player prevents the use of views of embedded YouTube content from influencing the viewer’s browsing experience on YouTube.

So they're not promising not to track users. They're saying they won't use their tracking to personalize anything.