With Fate/Stay Night, one problem is that I was looking at ripped videos on Youtube, while the original material is a "visual novel" with branching paths, so it's possible (but unlikely) that the people who put up the videos missed all the rationality-relevant bits.

I haven't watched the anime, but I have read the visual novel, and the anime does not have a reputation for being a very faithful adaptation. The visual novel at least does share themes that often feature in Eliezer's work, but I wouldn't call them "rationality content" as such. More in the manner of Heroic Responsibility and related concepts.