Theres nothing massively wrong with it, it's a perfectly plausible model.

The one thing that isn't particularly nice is that spheres have intrinsic curvature, essentially if you draw two parallel lines on a sphere they will eventually touch. We can go and look at astronomical data and see if the universe has any intrinsic curvature that we can see.

People did this and it turns out that from all the astronomical data we have the universe looks incredibly, spectacularly flat. No curvature at all that we can detect. This doesn't mean it isn't a sphere, but it means that if it is a sphere it's a really big one. Much much bigger than the observable universe.

To be pedantic, there is curvatures, see lensing effect of large masses. It's not global, though.

Ahhh, yes parts of that sound familiar. So, the reason we can (at least pretty much) ~~rule out~~ a 3-sphere [EDIT: I didn’t read carefully, and missed the “it could be a really really big 3-sphere so that the curvature is close enough to zero” part], is because you can’t have a flat sphere,

But you can have a flat torus (or some other shapes that “wrap around”), but we have different reasons to disbelieve those shapes (the “looks the same in any direction” and “looks the same in every position” expectations).

Cool, thanks!