So, for example, as far as I know, every web browser uses the HTML5 parsing algorithm for parsing HTML. This algorithm is very complicated, because it describes what parse tree to produce for any possible document. There's not, like, a separate HTML4 parsing algorithm; the HTML5 parsing algorithm replicates a lot of the complexities of pre-HTML5 parsing, but standardized.
Similarly, in CSS 2 the biggest complexities are things like floats, inline layout, margin collapsing, and stacking contexts. (My PhD was most of a standards-compliant CSS 2.1 layout engine.) There's not a CSS inline layout algorithm (the thing that puts words into lines) other than the CSS 2 one, though CSS 3 does add a lot more features.
In other words: the browser doesn't have a separate code path to handle old content. Instead, CSS 3 contains layout algorithms that extend CSS 2 but they don't replace them.[1] Similarly HTML 5. There are obsolete features you could probably dump (obsolete image formats, obsolete codecs) or rarely-used features (eval, document.write) or edge cases that probably don't matter that much (margin collapsing) or features where maybe your user base doesn't need it (writing direction? floats?) but this is really not so different from what the article talks about: a WebView/Electron replacement, where you commit to supporting a known universe of pages instead of the whole web.
[1] Granted some features like floats have become a lot less necessary, since a lot of their use cases can now be replaced by flex-box.
