When I press Ctrl-F, the cursor is positioned at the end of any existing text in the search box. Thus, pressing Ctrl-F and starting to type a search term merely appends the new search term to existing text, which is rarely if ever desired. Instead, I want Ctrl-F to select existing text so I can type over it. This is how Ctrl-F works in virtually all text editors (and web browsers, etc), afaik.
I know it is less handy with an extra step of Ctrl A.
Its counterpart Ctrl Shift F (all-notes search) does cursor in and prehighlight previous search terms. And an Esc does not clear the search terms (not the case for Ctrl F).