The only thing private mode is make it harder to follow your browsing history on your local device since it doesn't store pages in your browser history. You can still find them in temporary system caches. Your ISP and Google both still know where you are going and what you do.I am surprised the EU has not demanded a "select your default search engine" pop up yet upon set up.
I personally use Google Search only in private mode or Safari as a whole for that matter (not sure how much that brings actually to staying anonymous lol). The only annoying thing about searching in private mode is that damn cookie banner on Google Search. You actually have to scroll down to get the accept button and I always end up hitting the wrong button.
Anything but search is done via the Duck Duck Browser
Private mode has its uses, but just doesn't do what it implies.