Currently what you can do as a website owner is to use Google Adsense (https://en.wikipedia.org/wiki/Google_AdSense) and then you are kind of sharing revenue with Google but only when someone visits and clicks Google Ad on your website.
Nooo! This is not what I meant. I mean that embedding a search engine to your website without the user moving back to Google. So every website has a search engine for the entire internet. Not just Google. I want search to be kind of affiliate revenue to Amazon. So the developer embeds a search engine on the search page of a news website they get commission off the ads in a search result if the user decide to use search on the news paper's website. A search engine like Google for global search not just site-limited search.
Like do what Google is doing to websites but in reverse. Commoditize search. It'll require something more to get it right/product market fit but it's possible if you be creative.
Google is not the end product, its a compliment to websites. Websites need to commoditize Google instead of it being the other way around.
A search engine widget on website would probably look bad, would always be in different position and present a host of UX challenges which would add friction to adoption.
Every website is already able to search the internet via the URL bar of your browser nowadays. Its always there when you need it.