Re: Huh?
A "single site search" is relevant in that the query information is already known to be associated with the particular client. It is only known to a different site in a search for third party sites when the client follows the provided links and it is included in the referrer.
I am viewing the process from a step before the form being filled and returned to the server. I am viewing it from the blank form being presented to the browser. A search provider for third party sites concerned with privacy should really not set up the browser to fail.
The browser does not decide to use a GET method for the search form. The server provides the form with the GET method. The server is where it is known that third party links will be provided. So, yes a POST method search form is what the server needs to provide to the browser. GET and POST are not seamlessly interchangeable so the browser cannot just simply use POST instead, the server has to be prepared for a POST and the best way is to present a POST method at the time of form delivery to the client.
The browser can alter the referrer prior to making the request to the third party server.
Google search is using the GET method knowing full well that most links provided to the query will be third party and will result in a referrer with the query embedded. Leaving it up to the browser entirely to prevent the query from being seen by the link destination server.
The second part is the browser does have a role. It is permitted to change the referrer. There are rules it is supposed to follow and ones it should. Implementing those rules pertaining to sensitivity of information is challenging for the browser. The only way to be sure is to always strip the query portion from the referrer if the link is going to a different server or domain.
As far as I know, Google is still the search leader and now the browser leader.
Google is not the only search provider guilty of setting up the search result to include the query so a referrer with the query will be presented to a third party. Nor is Google the only browser provider guilty of sending the search query off to a third party by design.
Times have changed and it has been realised that more information is sensitive than previously thought. Google would prefer that we keep doing things the old way as long as possible.