Here is the secret to what you are doing, from my pulse and digital circuits class from many years ago. A square wave, like from your NE555 circuit contains a fundamental frequency, plus an infinite number of third and higher harmonics of your fundamental frequency. The squareness of the corners of the square wave, seen on an oscilloscope depends on the power in those odd harmonics. More power, the sharper the corners.
What you are listening to is a harmonic of your fundamental square wave frequency. You don't need to get the square wave to be the frequency you want to listen to, but can be 1/2, 1/3, or any other division. The power goes down as you divide down, but can still be heard on your receiver.
If the code signal, CW, you heard was a pure sine wave, you would not hear it on an AM receiver, no AM modulation. So, it is really a really bad square wave that makes it audible on an AM receiver.
Paul