How can this high-side IGBT drive work without exceeding maximum Vg-e
Voltage is the difference in potential between TWO points. Any two points, not just a point and ground.

You just drive the gate with a circuit that applies a gate voltage relative to the emitter, rather than ground. A so-called high-side gate drive, or floating gate-drive. Very commonly used in half-bridges.

That way, you don't have to worry about the load messing with the emitter voltage and causing Vge to get too high and blow the gate oxide layer, or worry about it getting to low and turning the IGBT/MOSFET off.

In this case, Vge would not actually be exceeded if you applied a ground-referenced gate voltage. What would happen is that current through the load would raise the emitter voltage to meet the gate voltage, reducing Vge and progressively turning the IGBT/MOSFET off. If you're lucky the IGBT/MOSFET will turn mostly off before it fries. If you are unlucky the IGBT/MOSFET will be half-on/half-off and dissipate large amounts of heat in the amplifier operating region and fry. The capacitor would never be able to charge fully and cause Vge to exceed its limits.