And the other way round: If the grid was large enough we would not need storage. It seems reasonable to try to improve both the grid and storage.
I still think everyone owning an electric car for storage is probably a more realistic plan, but it's definitely not impossible to create a world energy grid and run the world even just from solar.
Your comment is exactly right; a blend is what's needed. Though in the case of solar, a blend still requires very long, very high capacity HDVC cables.
In the long term, since the grid is a monopoly that is not eager to drop its price, whereas solar and batteries are likely to keep dropping in price, in part due to competition and in part due to appealing to early adopters, I expect that a few decades from now, the typical homeowner will use mostly solar and batteries instead of paying the monopolies' higher prices.
One might use excellent storage technology to, e.g., charge batteries in the Sahara and ship them to Switzerland (or, more realistically, synthesize oil in the Sahara and ship it Switzerland), but that's just a slightly more indirect way of addressing the problem of transportation- moving the energy from the place where it can be produced to the place where it is needed to be used.