Why FM is produced by varying the frequency of an oscillator in step with the audio. A reactance modulator places a voltage-variable reactance (a varactor or a transistor stage made to look like a capacitor or inductor) across the oscillator's tank circuit, so the audio voltage pulls the resonant frequency up and down. The deviation follows the audio amplitude and the rate follows the audio frequency, which is exactly frequency modulation. The low-level oscillator signal is then multiplied and amplified to the operating frequency.
Watch out A balanced modulator suppresses the carrier and outputs double sideband, the first step toward SSB, not FM. Modulating the final amplifier is how plate or collector amplitude modulation (AM) is done, not frequency modulation.
FM starts at the oscillator: reactance modulator changes frequency. Balanced modulator = SSB. Modulated final = AM.