This proposal won't do very much. Indexing is the (relatively) easy part of building a search engine. CommonCrawl already indexes the top 3B+ pages on the web and makes it freely available on AWS. It costs about $50 to grep over it, $800 or so to run a moderately complex Hadoop job.
(For comparison, when I was at Google nearly all research & new features were done on the top 4B pages, and the remaining 150B+ pages were only consulted if no results in the top 4B turned up. Difficulty of running a MapReduce over that corpus was actually a little harder than running a Hadoop job over CommonCrawl, because there's less documentation available.)
The comments here that PageRank is Google's secret sauce also aren't really true - Google hasn't used PageRank since 2006. The ones about the search & clickthrough data being important are closer, but I suspect that if you made those public you still wouldn't have an effective Google competitor.
The real reason Google's still on top is that consumer habits are hard to change, and once people have 20 years of practice solving a problem one way, most of them are not going to switch unless the alternative isn't just better, it's way, way better. Same reason I still buy Quilted Northern toilet paper despite knowing that it supports the Koch brothers and their abhorrent political views, or drink Coca-Cola despite knowing how unhealthy it is.
If you really want to open the search-engine space to competition, you'd have to break Google up and then forbid any of the baby-Googles from using the Google brand or google.com domain name. (Needless to say, you'd also need to get rid of Chrome & Toolbar integration.) Same with all the other monopolies that plague the American business landscape. Once you get to a certain age, the majority of the business value is in the brand, and so the only way to keep the monopoly from dominating its industry again is to take away the brand and distribute the productive capacity to successor companies on relatively even footing.