It’s better to think of the Big Bang as describing a point in time rather than a point in space.
What do you mean by that last claim? Any observable region is bigger at later times than it is at earlier times. The reason all points always appear to be moving away from all other points is that that is in fact happening.
What's the significance of claiming that the size of the infinite plane never changes? It's just as true that if you start with the unit interval [0, 1] and let it evolve under the transformation f(x) = tx, the size of the interval will never change -- every interval calculated at any point in time will be in perfect 1:1 correspondence with the original (except at t=0). But this doesn't mean that the measured length of the interval at different times isn't changing; it is.
https://www.wolframalpha.com/input?i=plot+%28x%2Cy%29+%3D+%2...
f(x,y) = (c * x, c * y)
f(x,y) = c * (x,y)
f(P) = c * P