Here are some comments from my limited understanding. (I invite Baluncore and other experts to correct me if I go astray.)
1) Piezoelectric operation for the tan headphones seems right. See, e.g.
Amazon earphones
They have an impedance of a few thousand ohms. Why not 8 ohms, which is standard for HiFi/stereo speakers and headphones? Because of the need to match the load impedance to the crystal rectifier. The broadcast AM band is, for discussion purposes, pretty close to DC so we can look at the DC impedance of crystals, which is typically a few hundred ohms (see, for example, the center column in this table from
Scaff and Ohl, Bell System Technical J., 1947).
View attachment 351422
These World War II microwave diodes literally used cat whiskers and semiconductor crystals inside of a cartridge.
Choosing the load impedance is a balance between best power transfer, which occurs with a matched load (500 ohms, say), and good selectivity (narrow bandwidth to capture just one station at a time) which requires high Q and therefore high impedance. I'm guessing that ~3000 ohms was empirically arrived at as a reasonable compromise.
EDIT: Another consideration is the need to forward bias the diode over at least part of the RF cycle in order to provide rectification. For a given power intercepted by the antenna, a higher circuit resistance will provide higher voltage across the diode. The forward drop sets the minimum voltage: 0.3V for germanium and 0.7V for silicon.
2) As a boy, I built a crystal set with enameled magnet wire wound on a toilet paper tube and held in place with airplane glue (toxic stuff, probably can't buy it anymore). The tube was nailed horizontally to a piece of wood and I used sandpaper to rub off the insulation along the top. A tin strip, nailed to the wood so it could pivot left to right and bent up to contact the coil, tuned the radio. I think this is easier than acquiring a ferrite antenna and variable capacitor, as you say.
3) I connected the RF input to a rain gutter downspout and the ground to a cold water pipe, and it worked! Gutters of old were steel with soldered connections, however, so this might not work so well with today's aluminum gutters and glued seams. Still, it's worth a try.
I think transistors are an obvious next step as the builder increases in ability to understand electronics (transistor action is not obvious...).