There are thousands of very simple FM transmitter circuits similar to this one on the internet that work. This one might not work:
1) The base voltage for the preamp transistor is 0.82V if the battery is new and/or the transistor has high current gain. Then its emitter and collector current are trying to be 1.6mA and the transistor is saturated and does not amplify anything.
2) The base voltage for the preamp transistor is 0.5V if the battery is old and/or the transistor has minimum current gain. Then the transistor is cutoff and does not amplify anything.
My FM transmitter uses a 5V regulator so that the supply voltage for the preamp does not change.
More problems with this one:
3) the oscillator transistor has a very low value 10k base resistor. Then the transistor is almost saturated and does not oscillate.
4) The emitter resistor value is high so the output from the oscillator (if it was biased correctly) is low.
My FM transmitter has a 47k base resistor, a 220 ohm emitter resistor and a 5V regulator for its oscillator.
More problems with this one:
5) The antenna is connected directly to the tuned LC circuit. Then the frequency changes if something moves toward or away from the antanna because then the capacitance of the tuned circuit changes.
My FM transmitter has an RF amplifier that increases the output power and isolates the tuned LC circuit from the antenna.
More problems with this one:
6) It is missing pre-emphasis (treble boost) that all FM radio stations have so the de-emphasis in all FM radios cuts high audio frequencies and makes the sound muffled.
My FM transmitter has pre-emphasis so that everything sounds crisp and clear.