I'm no energy markets analyst, so take this with a grain of salt, but I think the next major breakthrough for solar (not the slow, inevitable rollout we're seeing now) will be when somebody figures out an economical way export this periodic overproduction. There is basically never a time when humanity as a whole has an overproduction of free energy, but at present we also don't have a way to make money turning this surplus into a tradable product (like oil, coal, LNG, etc.) because all the electric-powered processes for making such products (ammonia, methane, primary aluminum production, etc) require big hunks of capital equipment that lose money unless they're operated more or less continuously. Battery, thermal, pumped hydro, etc. help here, in that enough of it can theoretically turn off-and-on solar/wind power into a continuous load to power your aluminum smelter or whatever.

Even better though, would be a cheap electrically operated methane plant that you could afford to run intermittently. This, plus a peaker natgas generating plant make, effectively, a battery of infinite size, or you can sell it to any of the many eager buyers of natgas.

Building a small, prefab, plant like this, if possible, would seem to be mainly a problem of scale, and therefore it seems likely that China will get to it pretty soon.