Could the spatially flat universe start small?
If this is correct, then there seems to be no tangible difference between the universe starting infinitely large or infinitely small
If you assume the universe is infinite, it has to be infinite at any given time. and only at the initial time, there is big bang singularity.
If the universe is infinite it was always infinite. At $t=10^{-10000000}$ it was still infinite. It has to be geometrically. But at $t=0$, we have a singularity.
You wrote that infinitely small means like a point, But the universe cannot be squeezed into a point, as you know. So even it's infinitely small it's still infinite.
If our observable universe started from a "point" in an infinitely large "space" and any other "point" is causally disconnected from us, then why do we need to consider these other "points" as "existing" in the first place?
Because the universe has to be infinite at any given time. So these points exist by mathematical definition.
What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe?
The universe started from a singularity. If its flat, it has to be infinite again. No matter how small it is.
If you mean "Why we cannot think our universe started like as an observable universe" my answer would be this.
1-Universe is the thing that encounters everything. So it still has to start from a singularity. And if its flat it has to be infinite
2- CMBR radiation shows that there is no preferred direction in the universe. Which points out that there cannot be any -away from point type- expansion. So even the observable universe seems to start from a point, It actually did not start from a point. Observable universe has no "real" center, There is just the universe and we have a limit on what we can see.
When you write "point" in scare quotes, what you're essentially doing is reinventing the notion of boundary constructions. A couple of good surveys on this topic are:
Sanchez, "Causal boundaries and holography on wave type spacetimes," http://arxiv.org/abs/0812.0243
Ashley, "Singularity theorems and the abstract boundary construction," https://digitalcollections.anu.edu.au/handle/1885/46055
The main thing to realize about boundary constructions in GR is that attempts to apply them to general spacetimes have failed. They are very convenient in the context of Penrose diagrams, but we don't have a useful general theory of them.
Your point about the nonfalsifiability of the existence of unobservable regions of spacetime is fine, but it has nothing to do with cosmology. You can take Minkowski space and do silly things like removing a point from it, or removing everything except for a certain region. This has no consequences for an observer whose past light cone avoids the missing points, but it's a silly thing to do, and we have no laws of physics that would help us to decide what the removed parts of the spacetime should be. This is why relativists only usually want to discuss maximal extensions of spacetimes.