That's the game, more sensitive receivers receive more noise too. The game is sending something that will not look like the noise. The longer the sequence, the more likely to decode something but the slower the symbol rate (bandwidth)

Imagine your array popped out SHORT-SHORT-SHORT-LONG-LONG-LONG-SHORT-SHORT-SHORT

You just heard a morse code for SOS! The shorts where detected over 100Mhz (FM) +30db for 1 second each and the longs were 3 seconds each on a carrier that sites at +10db. That's amplitude modulation and that looks like intelligence but unless you knew morse code - it wouldn't make any sense.

The further away you get from the source, the more those decibel spikes weaken and will eventually be no different than the noise floor. Your super computer with a billion ears, only hears ~static~.

Try this, imagine instead if there was no free space loss in the electromagnetic field - we'd wouldn't be hearing humming but SCREAMS from all the noise sources from EVERYWHERE as if we were right next to them, forever. It would impossible to decern anything from anywhere. Communication is defined by its distance because signals have differing origins. Sensitivity, or lack there in, is a feature not a failure.

Who's to say a quasar isn't just a lovely time clock for signals encoded in the noise and we haven't figured out what the breakpoint from noise is yet?