TBH I don't really know why I am wasting my time but I'll take the bait!
Firstly the Pave PAWS example of 'range' is not comparing like with like. The radar return needs to be orders of magnitude stronger to give information about (say) a missile - in the EME example, we're looking for typically a very narrow band CW (morse) or JT8 signal.
Here is a very rough path budget, showing that EME is very feasible...
Path loss at VHF is around 250 dB
US Ham power limit ~1.5Kw, around 60 dBm
An average receiver can detect -120 dBm
So -> 250 (path loss) - 120 (receiver detection) = 130 dB below noise ...
130 - 60 (power limit) = 70 dB below noise.
Antenna gain helps you twice (tranmit and receive) so assuming 20 dB gain (x2 =40)
70 dB - 40 = 30 dB below noise (you see, we are getting there!)
Use masthead preamplifier, typical gain 20 dB
30 - 20 = 10 dB below noise (we are *nearly* there)...
Finally, utilise sophisticated software and modulation schemes such as WSJT, which can recover signals from 20 to 30 dB below noise...
So - 10 - 20 = 10 dB ABOVE THE NOISE...
(I'm not an expert in this but the above is a reasonable approximation ... multiple stacked and phased antennas could give another 6 to 12 dB gain, and remember that antenna gain applies both to the TX and the RX signal, so CW is also possible and SSB (voice) is not out of the question!)