GPS is suprisingly low power. I believe the satellites themselves transmit between 20W and 50W, and in general the signal is quieter than the background noise threshold. It's only by correlating with the PRNG stream [1] that the data signal can be detected at all [2].
[1] The PRNG stream is 1023 bits at 1.023Mbps, so repeats every 1ms, and only autocorrelates with the correct stream when they are aligned. When the streams are not aligned, the data looks like random noise, and each transmitter has a different LFSR configuration to provide a different sequence such that each stream has a low level of correlation with another.
[2] The PRNG stream bits at 1.023Mbps are exclusive-or'd with the data stream at 50bps, so when the decoder is using the correct PRNG and sequence offset, exclusive-or'ing with that produces detectable long pulses at the expected 50bps.
No, conventional radio broadcasts can be received with a low noise amplifier and a tuned filter.
The received GPS signal, at ground level, is lower than the thermal noise floor. And the 1.023MHz code is modulated on the RF carrier anyway.
So correlating it with a sine wavelet?