If you don't mind all the terrible drawbacks of the approach, an AM radio isn't so hard to make. Especially a very simplistic and primitive one, like a spark-gap transmitter.
You can make an AM reciever using little more than a piezo button speaker, a reiostat, a ferrite core wrapped with enameled copper wire, and a diode. That's what basically comes inside a "crystal AM radio kit". The "crystal" is the diode. Traditionally, a germanium arsenide crystal.
Prior to the invention/discovery of semiconductor junctions, like a crystal diode, the main squeeze in radio equipment was the thermionic valve-- aka, the vaccuum tube. Vaccuum tubes are super simplistic to make, being at the simplest, a high voltage wire coil as the driven element, and a pickup cage as the receptor. The principle of operation is quite simple: electrons "bunch up" on the hot, driven element, and get "kicked off" by perturbations in the energy intensity applied to it. Literally, like an avalanche. Once it heats up, very small "flutters" in the applied voltage will result in an exagerated release of radiant electrons getting kicked off into the vaccuum of the tube. The collctor collects some of those electrons, and typically gets an "amplified" signal from what was originally supplied to the emitter as the input. A 2 pole vaccuum tube is basically just an extremely wasteful signal amplifier.
The AM radio would benefit more from a triode based design.
A triode is a little different. It has the driven emitter, and the collector, but it has a second, non-thermal pseudo-emitter grate between them. As analog signals get fed through the middle grate, it gets a momentary negative charge, but not enough to emit. Instead, it is enough to slightly repel the radiant electron emissions from the emitter so they don't get to the collector. This way, you can drive the emitter at a higher voltage, use the inverse part of the reference signal wave to control the grate, and have a much better fidelity on the amplified signal.
It will work better with an AM radio reception coil, because AM radios are little more than analog resonators with a tunable resonant frequency. The frequency of an AM broadcast is fixed. The amplitude (intensity) of the signal is what encodes the message. This is different than FM, which modulates the frequency within an allowed range of frequencies. Since the resonance of the AM radio coil is an analog wave frequency, you can use both parts of the wave (peak and trough) with a triode, where a simple tube amplifier would only use one part. (Just peak, or just trough.)
The issue with keeping an AM radio set working is well documented in WW1 radio operator's reports, and historic accounts of their use. Keeping the power hungry amplifiers well fed, broadcasting enough juice to cover any kind of respectable distance, dealing with fouling and breakage of the amplifiers, and a whole swath of other issues were non-trivial issues which greatly limited the effectiveness of the broadcast methods.
AM transmitters are basically quite similar to a reciever, but in reverse. This is why in the 1960s through the 1980s or so, "pocket AM broadcast" toys became quite commonplace after the invention of the semiconductor. Crystal oscillator circuits provide the reference frequency, which then gets amplitude modulated with the intended message. The now signal bearing wave is then fed through an amplifier, and into the tuned resonant coil-- aka, the antenna. Using vaccuum tubes the amplify the signal enough to broadcast big distances would require a very large, bulky, hot, and fragile transmitter, and since we wouldn't have access to a fast switching crystal oscillator for the signal reference, we would have to use very specially tuned wire solenloid "tank circuits" instead. The effective frequencies that can be reached with such circuits aren't nearly as good as with a good crystal oscillator though.
There are several nasty, filthy little caveats.
Without an oscilloscope, and a well made, and very tiny reostat, it is almost impossible to accurately and consistently build resonator tanks which operate at the specified frequency, due to defects in the wire, and other hassles. (A reostat is just a wire coil, that has a "positionable" pickup lead that allows the circuit to be tuned. Imagine a metal stylus being drug along a slinky, so that the length of the slinky involved in the circuit can be dynamically adjusted. There can be confusion with the term, because it can also be a variable resistance resistor-- like a carbon rod, with the metal needle being positionable down its length.)
Sorry, but I can't really play.