Daysleeper (or Daydreamer should that be) - sounds like you should be working for a large company in Edinburgh!
My simplistic understanding of radar - to detect sparrows need a short wavelength, short wavelengths have poor power transmission so need a big transmitter and big antennae which don't fit into the radome of a fighter aircraft. To detect stealth need radar returns above the noise floor and that given only a moderate power output in a fighter there isn't suficient returns until close range to detect a target above the ambient noise levels. Sure you can tweak all you want with wiggly amps, but the fact remains if you haven't got a massively powerful radar, a stealth aircraft (or a sparrow) will always be seen at short range.
So stealth is the way forward if you can't train a sparrow to carry and shoot AMRAAMs at the bad guys.
SG
PS - Time for some F4 blah about how they used to track Sparrows (Aim 7 variety) all the time!