The "Right to be forgotten" thing is murky too. If I lost to you in a competition and I"m deeply offended/embarrassed by this, is it within my right to remove your victories from the internet? Why have the people indexing the information remove it instead of the people hosting it? What if you mention the event again in an interview years after said removal? Are you in breach of some covenant for mentioning something-which-shall-not-be-remembered or is it the fault of the search engine for not maintaining some infinite list of embarrassments of humanity? Would said list of embarrassments be a breach of said right even though it would be the only way to enforce it?

Swap "run" with "regulated" and you'd have the same problem. A heavily regulated search engine would be a subpar experience. In some regard, it is already regulated in terms of what speech they can show. But that's not "search engine specific" regulation. Any specific regulations regarding search on the Internet has a high probability to degrade the utility of the service.

The "right to be forgotten" might just be a good example of this. Time will tell.