What happens to the windings on a transformer during tap changing?
When the load is increased, the voltage drops due to IR voltage drops in both the primary and the secondary.
If we ignore saturation, then the output voltage can be boosted back up to its proper level by modifying the turns ratio by either reducing primary turns or increasing secondary turns.
Doing all the compensation for varying load by tap-changing on the primary means that the flux has to increase, to compensate the IR voltage drop on the secondary. A transformer that is provided with only primary taps must be designed such that the flux is still below saturation even on the lowest primary tap with its highest input line voltage.
A technically better arrangement would be to provide a transformer with both primary and secondary taps. You would use the primary taps to control for line voltage variation, and both sets of taps to control for droop with load current. It's probably cheaper to provide one winding only with taps, and over-size the core slightly to provide headroom before saturation.