I was digging around a little, and it looks like the stylesheet is being loaded the same way whether the page gets styled or not. I looked at a few unstyled elements and ... the CSS (styling code) is ... interesting. ?
Whoever wrote the CSS made a lot of the styling dependant on which container it is in (so if someone shifted around the HTML, it would break the styling.)
It makes me think that someone (at XenForo) pushed a PHP change that is rendering the HTML (sometimes) in a way that it doesn't see the styling.
I would not want to be the person in charge of fixing this.