It doesn't necessarily have to be a direct payment. Companies can white-label the service or include it as a value-add as part of their offering; for example an ISP or mobile carrier offering this to their customers.
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.
