Or create a new partnership with various like minded organizations and create a new index that each can use in their own way (perhaps those you mention plus DuckDuckGo and others that my brain isn't bringing to mind right now [maybe Wikipedia, EFF, internet archive, maybe Nebius would help...]).

I know this wouldn't be easy but maybe such an arrangement would be possible where there is some shared index and then the partners can each use it, extend it... in their own way.

I could see a collaborative open source web index run by the WMF, Internet Archive, EFF, GNU, DuckDuckGo, and Kagi being too big to simply ignore like a previous commenter suggested might happen to a new index. I could also see Qwant being interested