They would, at best, be used to power small local induction forges and if you're lucky, refrigerators. There's no way you could use them to restart the grid.
Can common rooftop solar setups actually deliver power locally with the grid being down? I would imagine them taking at least frequency synchronization from the grid, and probably also not faring too well without the grid to compensate for variable output when a cloud passes over them etc.
Most rooftop solar can't produce power without the grid. They use cheap inverters that synchronize frequency from the grid. They also don't produce enough current to handle the peak loads from large appliances starting. The battery provides the current peaks. It is possible to buy more expensive off-grid inverters.
Yup. Stanford University discovered this to their dismay. Power grid went out, even with a field full of solar panels they had no electricity until the grid came back.
Typically solar will not feed back into the grid if it is down, to avoid energizing the lines and zapping repair crews. This is an intentional design, I’m sure anywhere with engineers or hackers could work around it.
Batteries can cost more than the solar cells themselves. Plus, the older lead-acid batteries required some maintenance. The easy path is to just stay grid tied and not bother with a battery.
If all the power lines on earth went down, I bet communities would connect people with panels to people with batteries, though, somehow or another.