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This is probably a really stupid question (yet to have my morning cup of earl grey), but which failing trick are you referring to?

There's a link called "web" on this page that links to Google:

https://www.google.com/search?q=%E2%80%98We%20have%20a%20fir...

It lets you into news sites that have some sort of paywall/registration requirement, but that whitelist Google.

Isn't that cloaking which is against Google's webmaster guidelines? https://support.google.com/webmasters/answer/66355?hl=en

[EDIT] Apparently they have added "First click free" policy https://support.google.com/news/publisher/answer/40543?hl=en

It appears to work inconsistently. I agree it's frustrating. It just worked for me, though it sometimes hasn't in the past. I haven't yet figured out the difference.

Can it be due to some browser privacy setting/extension interfering with (hidding) the referrer?

Not in my case. I haven't made any changes to my browser setup between attempts.

Right click the web link and open it in an incognito window (or your browser's equivalent).