Being idealistic about it won't change the outcome.
As far as I know, this hasn't really been tried, definitely not with the kinds of resources this project will have available.
PageRank was initially much less prone to SEO shenanigans because it relied on signals from other sites (incoming links) to decide how important a result was. Of course, as Google became more popular, people started sharing links on other pages and so on to cheat the PageRank algorithm. And Google have been caught in a fight with SEO ever since.
AFAIK, nobody has ever tried that because of the dangers of vote buying and coercion. Essentially, gaming the system.
(I don't think this proves anything! Simply wanted to suggest that comparing search to democracy doesn't significantly change the analysis wrt opacity.)
That's a major fallacy you're opening with. If something is important to us (as a society), we will find ways to make it serve us well even when it's under attack. The rest of your post even makes a similar point: the social value of well-working search is greater than the economic value for even the biggest search monopolist on the planet. So why not socialize it?