I propose to create an add-on for Chromium browsers (Chrome, Opera, Brave ...), which would adds kagi to the list of search engines, transfer our session id to the private browser mode and when we enter the private tab, it switches the search engine to the private mode.
e.g. you .com (https://chrome.google.com/webstore/detail/youcom/afiglppdonkdbkkaghbnpklddbemkbpj/)
