I think it's a logical necessity for the universe to be infinite, at least spatially. In order to be finite, it needs to have something outside of it to set a boundary containing it. But if that were the case, then there must be something beyond it, which means the space contained within that boundary would not be the entirety of the universe.

Whether that space beyond is filled with anything or simply empty until stuff expands into it is a different question. And whether there were multiple other big bangs incomprehensibly far away from the observable universe is another question too. But neither of those possibilities implies that the big bang would have happened at every point in space simultaneously.

Another reason that possibility is untenable is because of heat dissipation. If every point in space exploded simultaneously, not only would there be nowhere for the force to go, but there would be nowhere for the heat to dissipate too, either. The heat would be uniformly distributed throughout space, offering no possibility of cooling down and coalescing into denser states of matter. The pressure would also be infinite, with no gradient. Everything would simply be an ocean of gammawaves, with no room for expansion.