Actually the Results List already persists across clicks, but there is currently no way to index through results (like a prev/next result button, handler or keyboard shortcut) except manually.
This video shows Find failing since discourse uses Lazy/Virtual lists, so we switch to Search within this topic, but there is no easy way to navigate between results without reopening the search menu, which cover the page, so it must be dismissed to scan over a result, then re-opened, scroll again, manually find the last result we selected, and click the result after that.
It is made worse since the search results cover the page so they must be dismissed to read a post and reopened every time.
If we’re going to replace the browser native Find function, which has existed in its current state since time immemorial, it should be at least equivalent in function.
The list is there, computer are great at moving through lists, and I can’t see anything complicated about it.