Hello and thanks for the site improvements!
I'm not sure if the skin is really the problem here.
On website pages, the information was previously distributed in tabs, but the tabs are not rendered anymore. Instead, the tab contents are placed unstyled one after the other vertically (which is a good fallback). This may be some broken extension or script. Or maybe templates like https://wikiapiary.com/wiki/Template:Website were (ab)using HTML elements that the foreground skin was automatically converting to tabs, but changes in the skin made it no longer work...
Looking at the developer console of my browser, I see some requests that lead to 404 errors, for example: https://wikiapiary.com/apiary/graph/graph.js, although I'm not sure if that has anything to do with tabs, though.
I've seen WikiApiary has Extension:Header Tabs installed, but it doesn't seem to be using it. I've updated the Template:Website/sandbox https://wikiapiary.com/wiki/Template:Website/sandbox to use those tabs and they are functional. We can switch to this use of tabs that looks easier than trying to fix what was being used previously to render those tabs.
Best regards,
-- Jesús Martínez Ciencia Al Poder