How not to check if Varnish is working.

Our new Varnish clients who choose to install everything themselves want a way to verify if it's working - and often fall in the same trap: they use the isvarnishworking.com website which has good SEO and promises to be an easy way to check if everything is running correctly.
Most often the answer is "Nope" - indicating that Varnish isn't operating - even thought the cache is running. I checked I can work around that - maybe they're doing some simple check that can be passed with a minor modification to the VCL file.
Unfortunately, I've found out that the data presented on this website is completely useless - the headers they show to visitors are different from actual headers sent by the server!
You can see that yourself below:

The headers isvarnishworking.com shows

Actual headers, as seen by wget

I don't understand what's the point of building a service showing fake HTTP headers.

When testing, just use the Network tab in developer tools in your browser, or wget/wbox if you're a command line user.