If I was a fingerprinting company, I'd be cross-referencing signals between browsers for sure.

If the browser header says windows but the fonts available says linux, that's a very distinctive signal.

And if the UA says Chrome but some other signal says not-chrome, that's very distinctive as well.

The article also mentions this, and suggests the UA is not a silver bullet. That said, they didn’t go into specifics. I’m assuming there are other details that correlate to particular browsers that will betray a false UA. Plus, having a UA that says Chrome while including an extension that’s exclusive to Safari (tor example) will not only contradict the UA, but it will also be a highly distinctive datapoint for fingerprinting, in and of itself.

You can change the header, but browser developers are not that dumb and they added properties like "navigator.platform" which do not change and immediately give you away. Consider also writing a browser extension to patch these properties. Also, I think that DRM module (widewine), that is bundled with browsers, also can report the actual software version. Sadly it is undocumented so I don't know what information it can provide, but I notice warnings from Firefox about attempts to use DRM on various sites like Yandex Market.