> The big bang didn't happen at a single point, it happened everywhere.

A bit of an odd thing to say, since "everywhere" implies there are multiple places to be, and at the instant of the big bang, there was only place to be. So it was both everywhere and at a single point: it was at all of the single place there was to be.

But your point (sorry) about expansion being from everywhere isn't specific to the big bang; space was expanding well after the big bang and it doesn't seem like the expansion ever had a "center" (at least, not since not-center places existed). It's expanding from everywhere. (But evenly everywhere? I have no idea. Hey, maybe black holes are like buttons in cloth, and it expanded everywhere except for where the buttons were holding things still at a rate relative to distance from the button. A brilliant hypothesis that explains exactly zero unexplained phenomena, at least none that I know of.)