I love this but the act of indexing the web is both complex and ever-evolving (not to mention expensive), so the market needs to have natural incentives for the public crawler to continue being good at its job. Google is incentivized to have a great index because it feeds into its competitive advantage. If the index is a shared resource, what incentive does the indexing company have to keep improving it? And what guarantees that the indexing company's strategy doesn't end up being unfair to those affected by their websites getting indexed?
All search companies would have to maintain secondary indices to augment the public one. There's probably some artificial structure needed to make open indexing viable.