Add to those first 3 bullet points of how you can use grid power when a battery doesn't run out . . .
Inverter limitation:
If you have an inverter of lower KW rating, say a common 5kw inverter with 6.6kw of solar and whatever battery, and you draw 6kw for an hour, your inverter can only put it's limit into the home's usage at any one time, in such a case you would draw 1kw for that hour from the grid.