One of the design rules mentioned in the article is that they don't rely on the home page. Instead they assume you get there from Google.

I just tried "renew uk driving license" in Google and got the right page as the first result. Same when I wanted to renew a passport recently, and for the birth example you mentioned.

There used to be one downside to this: scammers would sometimes get search results above the gov.uk one with sites that were not-quite-claiming-to-be the official site, and charge you extra for the "service" of applying on your behalf. However that seems to have been stopped now. Maybe HMG have had a word with Google about it.