A publicly owned asset doesn't need to be managed by a private entity. The incentive can be exactly the same incentive that the NHS and NASA operate on. It's a fun problem that you can make a career on making better that is ultimately a public good. Not everything needs to be operated with pure profits in mind.
> And what guarantees that the indexing company's strategy doesn't end up being unfair to those affected by their websites getting indexed?
Voting/lobbying/etc. Though I'd assume everyone that makes a website will want to be indexed.