Create a link with a query to search on the user chosen default search engine

Today if a website wants to add a link that opens a search engine, it needs to pick one of bing, baidu, duckduckgo, google, or others.

What if the website wants to open the user defined search engine instead? Most browsers provide a search mechanism in the address bar today and it has become the default (to be confirmed)

Proposal:

Even though there's plenty of explanation
for <a href="search://?q=what does the fox say?" target="_blank" rel="noreferrer">What Does The Fox Say</a>,
today you are really going to hear what does it have to say.

A polyfill as a fallback could be to use the website tailored search engine url, but most browsers for privacy sake doesn't allow JavaScript to detect what is the search engine of the current browser session.

You can find a relatable behavior described with x-web-search:// at https://www.macstories.net/tutorials/launch-a-web-search-in-safari-from-other-ios-apps/

Start by reviewing the proposed search://?q=... URL behavior in the issue and the related x-web-search:// example. Compare the proposal with how browsers expose address-bar search and consider the privacy implications described. Done requires an agreed HTML Standard behavior and a clear specification of how user-defined search engines would handle the link.