The zigzag config was done a lot at one time, it was common for a generator to have 12 windings, these could be arranged to give 55-0-55, 230, or 380 volt phase to phase.
The use of just one phase, would depend on which phase the regulator monitors. That is if you have a regulator, control varies a lot, I have worked on generators which used voltage and current to power the field windings, and at a set output the voltage was spot on, but under or over that output, it could vary a lot.
Today we see a lot of inverter generators, the output voltage and frequency may vary a lot, but the inverter corrects it to 50 Hz and 230 volts. This allows a generator to run at reduced revs with a light load.
So a 1 kVA generator may not work in the same way as a 250 kVA generator, and no one can give an answer without knowing how the generator being talked about is controlled, I have worked with generators which were really car alternators converter, to 750 MW. And they don't work in the same way.
So I can't comment on Honda etc, but some idea of the internal wiring would help.
Of course, you could feed all into an inverter and get a smooth AC output.