We take for granted that browsers have a default search engine, that users are incapable of using another search engine, for example adding "reddit" to their google queries rather then using reddit's search1, and that the default search engine funds the browser's development.

What kind of UI could make it easier to use other search engines?

tab-to-search: just as you can press enter to go to the suggested website in the addressbar, you can press tab to search it. (this feature already existed in chrome, but now has to be manually actived for every single website.)
search with different search engines at once. this could mean opening different tabs with the results from different search engines, opening one tab with different columns for different search engines or aggregating the results.

How could a browser that implemented such features be funded, which companies would have an incentive to support it?

big companies with search functions that wouldn't work as the default search engine, for example amazon, twitter and reddit.
companies that provide search engines to other websites, such as algolia and typesense.

Any website that has a search function would have some incentive to promote such a browser.

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That hack only works for information that's available on the web, not for your emails, messages, notes …

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