Maybe taking solar into account, or maybe as an average, but there's no way you use just 22kWh in a year.

Do you understand the difference between AC and DC? Your panels and your batteries produce DC power. The inverter converts that to 120V 60hz.If you were standalone solar, your inverter can use its own 60hz clock and it would be fine. But you tie into the grid, so the 60hz current produced by the inverter has to be exactly in phase with the power supplied by the grid. Otherwise, any power back fed would just be distortion and would end up dissipated as heat. So the inverter shuts down when it can't sense grid power. That's also a safety feature, so that your system isn't feeding back to the line when there is a blackout. Imagine how that would work if there are people working on the lines.

If you add a generator, the inverter should once again work, because the generator is producing 60hz current which can clock the inverter. There will be a transfer switch that isolates the house from the grid, so there is no phasing issue. At that point there's probably not much point in adding back the solar panels, because a) solar power can't back feed the generator and b) it won't have a material effect on generator fuel consumption.