Maybe the double delta doesn't work, but if it doesn't work why have the generator makers gotten away from the zigsag in favor of the double delta?
The zigzag will put equal load on all of the windings on both the the primary and secondary when you are supplying a 240 volt single phase load. One of the three coils directly supplies 120 volts or half of the 240, they other 2 connected so that 120 + 120 at 120° =120 and that supplies the other have of the 240.
Again, I agree that both the zig-zag and the double delta _use_ all of the three phase coils in the generator, and that such a configuration would similarly use all of the coils of a transformer. I also agree with your phase angle math describing how the terminal voltages are produced.
My point is that the current flow though the 'angled' coils will be in phase with the _single phase_ output voltage. The same current has to pass through all coils in series.
The power factor for the 'inline' coil is the expected single phase power factor. The power factor for one of the angled coils leads this by 60 degrees, the power factor for the other lags. All of the coils see the same full current, and all are utilized. Using the zig-zag or double delta arrangement clearly improves the alternator power handing capacity for single phase loads.
When the source of power is a spinning field moved by an engine, the engine sees a pulsing mechanical load (smoothed by inertia).
When the source of power is a set of primary coils, you see the same phase angle and power factor effects. The net result is that current flows in all of the coils. But what changes is the supply current going into the primary terminals.
In the transformer setup, you have 12 different currents: the 3 phase supply currents, the currents in the three primary coils, the currents in the three secondary coils, and the 3 load currents.
With single phase loading one of the load currents is 0.
With the zig-zag or double delta connection, all of the primary and secondary coils are carrying current. It is to utilize all of the coils that the generator guys use the double delta arrangement.
I claim that in the double delta transformer setup, one of the _primary_ supply currents will be close to zero.
-Jon