I never knew that, though I knew the airflow was somehow varied through the turbine. I thought the exhaust area was simply altered to vary the speed of the flow.
Pratt & Whitney tried that with the R-2800 and R-4360 VDT (Variable Discharge Turbines). DIdn't work terribly well, and probably needed FADEC to work properly. One R-4360 VDT was flown in a B-50 - the flight engineer had to constantly monitor the engine and adjust the nozzle.
The wastegate was the method used to control exhaust flow through the turbine, excess exhaust bypassing the turbine. And it is still done to this day - almost every turbo engine on the road has a wastegate.
The turbo engines in F1 have a different solution - they have a turbine that is big enough to cope with the entire exhaust and use a motor/generator unit to control the speed of the turbo. It is a turbo-compound. They still have a wastegate to dump the exhaust for short periods of extra power (the motor/generator is used to drive teh turbo in these situations).
In the Porsche 919 Le Mans car the turbo's wastegate feeds into a power recovery turbine, which is connected to a generator.
In WW2 era engines the Wright R-3350 had a turbo-compound version post war. It had three turbines, but these were not connected to compressors, but were mechanically linked to the crankshaft (with fluid couplings).
Allison also developed the V-1710-127 (-E27) which used the turbine from the C-series turbo (as used in P-47s) to feed power back to the crankshaft.
In these two cases it was the mechanical link which prevented the turbine overspeeding, so a wastegate was not required.