Google site search is on the way out. Now what?
https://searchengineland.com/google-site-search-way-now-271366
Had problems and stop using it after seeing this type of news?
That article is about something else, namely Google Site Search, a paid service by which a site could farm out its internal search function to Google. No ads. No Google branding.
BTW, as your link points out, Google Custom Search is still available and can be used for some of the same purposes as Site Search.
If I were webmaster of example.com, I'd be tempted take whatever the user typed into my search box, stick "site:example.com" in front of it, and open Google in an IFRAME. Of course, the result would contain ads, but so what?