My 'bad' mental model is: In the beginning there was an infinite amount of space and somewhere within that space the Big Bang happened. The universe is all the matter and energy produced by the Big Bang and that universe is expanding at a maximum speed of C through the infinite amount of space.
But you are saying, I think, that before the Big Bang there was no space. The Big Bang produced all the matter, energy, and space itself. And space is expanding, NOT just the matter and energy expanding through the infinite pre-existing space that I imagined.
I still have a very hard time imagining 'no space' before the Big Bang and the concept of space being something besides 'nothing'. Because if space is expanding it must be something more than nothing.