In the first circuit, the 4.7k value of R2 that biases the transistor is very low and the 470 ohms value of R1 the emitter resistor is fairly high so I think the transistor will be saturated and will not oscillate.
For a regulated 5V supply, my FM transmitter uses 47k for R2 and 220 ohms for R1. The inductor and RF capacitors are the same as this simple circuit. It has a 5V low-dropout voltage regulator so that the frequency doesn't change and so that the preamp still works as the battery voltage runs down. The preamp has gain and pre-emphasis (treble frequencies boost) like real FM radio stations so that the de-emphasis (treble frequencies and hiss cut) in all FM radios does not cause the sound to be muffled like in this simple circuit shown here.
EDIT: FM modulation fluctuates the radio frequency with the audio. The audio causes the transistor to fluctuate its current then its capacitance also fluctuates which fluctuates the frequency producing FM. The circuit also produces amplitude fluctuations (AM) but FM radios ignor AM.