Why A balanced modulator suppresses the carrier but leaves both sidebands (double sideband suppressed carrier). To get single sideband you must remove one of them, and that is done with a sharp bandpass filter, typically a crystal or mechanical filter a few kHz wide placed at the IF. Shifting the carrier oscillator to the other side of the filter passband is how the rig swaps between USB and LSB.
Watch out The carrier oscillator feeds the balanced modulator and sets which sideband lands in the filter passband, but it does not itself do the selecting; IF and RF amplifiers only add gain, not selectivity of that sharpness.
Balanced modulator kills the carrier, the filter kills the extra sideband.