Why do higher fluorine oxyanions not exist?
There are two factors at play here:


Fluorine is more electronegative than oxygen. In oxyanions, the central atom carries a substantial positive charge due to the oxygen atom pulling away electron density. In oxyfluorides, oxygen actually donates electron density to fluorine and oxygen would logically be the central atom. There are theoretical studies into $\ce{OF3+}$ but no successful syntheses yet.
The fluorine atom is tiny, even by period 2 standards. There's simply not enough space around the proximity of the atom to fit three or more other atoms.