Change search engine in search bar on home screen on Android 11?

When I set up my new Fairphone 4 with Android 11, I selected DuckDuckGo as my search engine. This is now used in the bottom search bar on the home screen. Now I can't find the place in the settings to change this to something else in the settings. And searching the settings for "search" brings up nothing :-/ Where can I change the search engine for the search bar on the home screen? This is what I need to find again: https://www.android.com/choicescreen/ The EU forces them to provide me with a choicescreen so that I can decide freely which search engine to use by default. But like for cookie banners, the EU guys are not the brightest when trying to control Megacorps, so they seem to allow Google to hide that screen very well. All descriptions I found refer to changing the search engine in a browser only. (It is lamented elsewhere already what a pita it is to have the search bar down there, but it seems we cannot get rid of it. In case someone knows better, that would suit me as well.)