You should be able to make a simple regen receiver ... have an insulated copper wire maybe 20 cm long ...
Well, that's an interesting concept. But have you ever tried a regenerative receiver? The difference between maximum sensitivity and oscillation is a very fine line indeed. A misadjusted (that is, oscillating) regen RX is a transmitter, that would be even easier to detect than a superhet. Back in the early days of radio, regenerative receivers were notorious for transmitting a signal that generated an interfering heterodyne in nearby neighbors' receivers.
And I'm not sure that such a short antenna would be sufficient.
EDIT: And separate controls for regeneration and tuning are a must.
Maybe you have some other ideas, such as an RF amplifier (which I have never seen in any regen schematic) with good isolation. But if you can build such a regen receiver, you're a far better electronics technician than I am. :-)