There is another.. Regulatory problem.. the EU, GB (I believe and the US to name a few have laws about the allowable RF power that a human being can be safely exposed to and it's in the crystal Radio class... A high efficiency red led will show some light @ 1V8 and 500 or less uA or ~9 mW. There is also the thought that the human body is considerably larger and can absorb a lot more power. I know of three radio tech's that are now deceased.. all about the same age and all of brain tumors.
BTW Crystal headphones didn't become popular until the mid to late 50's. They were the really cheap replacement for a much more expensive high resistance type of magnetic earphone that was very sensitive. Brush, I think were the best. Second best was the receiver in a telephone of that era.
Bob