Generating an AM signal is child's play compared to detecting one.
To generate a signal you just need to create an oscillation and vary the amplitude, which can very crudely be done with PWM.
To receive a signal you first need to isolate that signal. There's a huge number of signals and noise around all the time, and you need to tune in to the signal that you are interested in. That's what you are doing when you twiddle the dial on a radio. You're selecting which frequency to listen to.
And to do that requires a tuned circuit - a kind of band-pass filter which isolates and amplifies the frequency you are interested in.
Imagine you are in a large hall filled with people. Every single person there is just standing around talking out loud - everyone is saying something different. Some are even just screaming, or whimpering in the corner. It's just a cacophony of noise. Impossible to know what is going on. That's what the air waves are like. You need some way of "tuning in" to one person's voice so you can understand what they are saying. As humans we often do that using sight, to watch a person's lips and our clever brains link the movement to the sound and help us pick out their voice. Coupled with our ability to select on the tonal qualities of a voice as well, we can pick out a single voice in the noise. That's what the tuned circuit does. It picks out one voice from the mayhem.
The Arduino can't do that. It would need to use some fairly sophisticated filtering algorithms, and it just doesn't have the power for that. Firstly you'd need to sample the raw RF noise at a minimum of twice the frequency you are interested in (so multi-mega-samples per second), and then you'd need to do digital filtering to isolate that one frequency. From then you'd need to do more filtering to demodulate the original signal from the modulated signal.
It's all possible to do in software, but not with an Arduino. It takes a good ADC and a DSP to do all the hard work.
Easier to just build a small tuned circuit from a few passive components.