Can this FM detector even work?

[Tom Albrecht]

That's a strange circuit, but may be functional as a "slope detector." That means it is simply an AM detector, but the resonance of the tuned circuit is tuned slightly off the IF peak. That way, as the frequency of the carrier is modulated, it moves closer to and farther away from the tuned circuit resonance, causing the amplitude to rise and fall.

Normally you don't need two diodes for a slope detector, and I'm not quite sure what is being accomplished by having two diodes here.