If you amplify an AM signal with a non-linear amp it will demodulate the signal. Some AM detectors operate in the non linear region and are square law or small signal detectors.
If you rectify an AM signal it will be detected. A detector operating in this region is a large signal or linear detector.
A simple crystal detector is a small signal detector and operates square law. This is not an advantage it is just the way it is. As the signal increases the output increases at a faster rate because of the non linear detector but it is still weak.
A receiver with an RF amplifier will boost the signal to the point it operates large scale or linear. It will have a higher output because it was amplified not because of the detector operation.
If you build an amp attach an antenna to the input and hear audio on the output you have a non linear amp or you or over driving it.
non linear = square law detector
over driving = linear detector.
If you connect the antenna and hear AC it is just line noise. When adding a diode across the input produces audio it is high gain enough to trace signals in a radio and linear enough to give valid test.
If your amp passes these test it is ready to serve as a signal tracer.