In a browser, you can easily start separate searches that are tied to that window, leave that search active, switch to another tab or window, and start or resume a search there. CTRLF obviously can't do this, but also doesn't report anything if you try (Ctrl-s remain silent fails if a search is running in another window elsewhere). ctrlf-forward-symbol-at-point on another window/frame during a ctrlf search does do something: it deletes everything else in the buffer except the symbol at point!
What should ctrlf do? Perhaps follow isearch, and silently abandon the pre-existing search and just start a new one in the current buffer.