APU uses either battery only or combination of battery plus AC to start. Start is much quicker than on battery only.
To answer the original question - if you don't start the APU then you're in a world of pain if the remaining engine generator goes offline, and you'll still need to start the APU. Then if it doesn't work you're back in the exact situation you wanted to avoid in the first place.
Starting the APU means that you're in a much happier place from an electrical perspective. I'm convinced some people come up with these crazy theories just to get a name for themselves.
On top of that the APU is much more likely to start successfully if you have both AC and battery power available, as opposed to trying to start it with the battery simultaneously supplying the entire DC ESS network.