How does this radio transmitter circuit oscillate?
Yes, it could oscillate, but in a SPICE simulator, it didn't. Not quite. A few component changes did start oscillations. The 7MHz crystal equivalent circuit is a guess (C1, L2, R5, C2):


The base-to-emitter capacitance of the 2N2222 is large enough that this is a Colpitts-type oscillator.