Can I make a homemade radio transmitter?
Your circuit can actually oscillate, and might even oscillate at a frequency within the FM band. It is a minimum-parts-count circuit that can fail to oscillate from too many causes. For example, the battery is in part of the circuit that is oscillating, and its stray coupling to adjacent parts will greatly affect frequency. It is not a circuit that will yield repeatable results.
Even if oscillating, and oscillating at the correct frequency, it will not produce your desired tone in an FM receiver. It will merely produce silence. You need an oscillator with provision for frequency or phase modulation, and an audio oscillator to provide that modulation. an example tone transmitter

A transmission-line oscillator is shown below that uses no coils or small-value capacitors. Frequency is set by the length of twisted-pair transmission line: a one-quarter wavelength is desired ideally, but a shorter length yields inductance that combines with gate capacitance for resonance. You might start with a length slightly longer, so that you can trim the length (and re-solder the ends) to raise frequency. For FM broadcast band, a length of about 15 inches might be a good starting point (its been awhile, my memory of lengths &amp; frequencies is uncertain). DC supply voltage to the chip is not shown. Use no more than five volts, and include a 0.01uF capacitor with very short leads from supply pin to ground pin.
This is a single-frequency oscillator - I have not tried to frequency-modulate it, but superimposing an audio-frequency signal on the Vdd supply voltage should result in FM.