I used to use a search engine called Searx (I was using this instance), and they had an option to send searches via POST request. This removes the search query from the URL parameters, making it so that searches do not leak in your browser history.
For example, instead of https://kagi.com/search?q=kagi it would just be https://kagi.com/search.
Their settings UI looks like this, and is set to POST by default:

It would be an optional setting, that users could toggle off and on.
They also have a setting to remove the search query from the page title, which would complete this setup.