Where should the profit come from for a search-oriented product/business?
Imagine that every Internet user pays just one dollar a month. That's still billions worth of revenue, and it can be segmented further such as basic features available for free and more advanced users (such as large-scale API access) can pay more.
At the moment I'm not aware of any company whose primary service is search. I believe the demand would be there - OS manufacturers currently have to either put up with Google (even if their goals diverge - for example it doesn't work well with Apple's privacy-focused marketing) or Bing.
A third player whose primary business is search (as opposed to using it as a loss-leader to lure people into their ecosystems, like Google and Bing are) could get some use, not to mention that programmatic access to large web index would no doubt be a service many companies would pay for. Think Algolia or hosted ElasticSearch but pre-seeded with a broad index of the web.