I'm just learning this stuff so the details are hazy, bu my understanding is that there's a difference between spacetime curvature and spatial curvature. You can have a hyperbolic spacetime while at the same time having a flat three-dimensional spatial section of it.

It's not an assumption that space is flat. GR doesn't specify the global space curvature, so it's possible that it has a globally negative or positive curvature, but so far there's no evidence of any.