Your assumption is mostly correct - “day side power can be used on the night side”. Say that you live in a city where the sun sets at 7pm. The largest synchronous power grid in the world is in Continental Europe - from east to west, it’s approximately 5600 km, and connects Portugal (UTC 0) all the way to Turkey (UTC +3), covering three time zones. That means when the sun sets in Portugal, solar panels in Turkey are still generating power at 75% efficiency.
As you can understand, this is an entirely different claim from “we would get power from the other side of the world”. It’s a strawman because that’s the weakest possible version of the argument I made.
As for the US power grid, no, you’re right, I was totally wrong about that. I had thought that the east and west power grids were connected, but it seems that they still haven’t sorted that out yet. Thanks for correcting me. It looks like they have a project in the planning stage to make it happen (Tres Amigas SuperStation) but it probably won’t be for a while. It’s absolutely achievable, though, and pretty easily.
It would also be achievable to get a single planetary power grid, theoretically, but I think it’s practically impossible to achieve that at the moment. It would need a level of global cooperation far beyond what we have ever accomplished. Definitely a future goal for our species!