That header is the user-agent's decision, I consider it an anti-user feature. Google (like most search engines) wants to know what you clicked on.

That's correct. Google uses Referrer-Policy [1] to prevent the full referrer from being sent to the destination website. The transition page is purely for tracking. The ping attribute [2] is meant to replace that transition page eventually, but it needs to be ubiquitous and equivalent in terms of features before Google switches to that method.

[1]: https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Re...

[2]: https://developer.mozilla.org/en-US/docs/Web/HTML/Element/a#...

I vaguely recall the `rel` attribute value "noreferrer" on the anchor element producing this behavior; is that not the case (any more)?

That's graduated into Referrer Policy, more or less, but it should still work.