OK, I purposely didn't post immediately when I saw your pretty pictures, tried to give it a day of thought... and I'm still not sure what to think about them. So my post may still sound more like "thinking aloud" than anything conclusive.
Well, first of all, please oh please don't turn into a zephir

Other than that, I still don't see what the second image is trying to represent at all... it only shows the surface of the sphere, so doesn't give an idea about how the size of "pieces" scales with the distance from the center...
But I guess I understand your idea - it
doesn't scale... it has the same size "pieces" everywhere, just more of them on the outside.
More "pieces" just means more space, greater volume, longer lengths for the circles the larger the radius is; a circle is a circle and an arc is an arc, not a straight line through curved space or some such junk;
This is just the normal geometry we all know and love, am I right? We just have normal flat euclidean space (maybe ignoring some tiny local curvatures caused by matter), and matter is bunched up in it in the shape of a sphere... you are right then, if we are not in the center, we should see much more stuff towards the center... the closer we are to the edge the more obvious the differences should be...
I don't think we've seen anything like that though.
Now then, what could the first image represent? I have several possible interpretations...
One possibility is that each "piece" represents the same amount of space as measured from inside that "universe"... each line segment is the same length, and is a
straight line, despite it being curved on the image.
But I have two "gripes" about this interpretation...
First, if all of these lines are considered straight, all of the drawn "corners" are right angles, etc, how is the "curvature" on the drawing "obvious" at all? How is it relevant, and what does it represent? Isn't the presented geometry in fact just "flat" space, but looping in two dimensions and bounded in the third dimension? It can just as well be represented by a wide and short parallelepiped, with the clarification that the top and bottom bases are actual bounds or edges, while the vertical surfaces around it are just a "warp portal" or something, linking each point on them with the same point on the opposite side.
Sorry, I'm not into pictures... maybe I'd draw you some if you say you don't understand me

Second, why is the "center" missing from the picture? Is it possible to include it, and what would the picture really look like in it?
If the tangential segments become larger the further from the center they are, shouldn't the radial segments do so too? This seems like a minor gripe at first thought, but I think it is important when we try to imagine how the image would look closer to the center.
With ever decreasing segments the closer we get to the center, we will essentially have infinite many segments before we reach it, so effectively it is just a point infinitely far... This version of the picture I think would be equivalent to the "bounded/warping flat space" I described above, just with one of the bounds removed... essentially, a space looping in two dimensions, and in the third dimension it is infinite in one direction and bounded in the other direction... a parallelepiped with a bottom base, but without a top, going infinitely high instead, and again with the mentioned "warp" sides.
If on the other hand the radial segments do not become shorted with approaching the center... I'm not sure at all what we will have.. somehow perfectly parallel lines happen to join into a single point, within a finite distance... not a geometry I can imagine. Infinite curvature, or something?
Well, now that I think about it, the square with "warp sides" thing that I described won't work exactly the same as "surface of a sphere", as a large-scale measurement from inside would be able to determine exactly in which directions it is aligned... a flat circle or cylinder with "warp circumference" on the other hand would tend to distort objects passing through it... so some kind of "surface of a sphere" looping of space is probably the only accurate description...
You are correct then, such a curvature will be indeed obvious, or at least potentially detectable in "large scale" measurements, meaning on the same order as the length of the loop. But in fact a smaller radius means a bigger curvature, so I think the "obvious stretching" would be towards the center, not "towards the ends". The strange thing that happens with parallel lines in the center itself is actually "infinite curvature"... singularity perhaps?
I don't see why we'd have that looping happen in only one or two dimensions and not all three, and I don't see why its curvature (or the distance from a "center") would be related to the third dimension... it is just not symmetric, I prefer to think that a realistic model will have all spatial dimensions be equivalent and indistinguishable.
And now a completely different interpretation of the picture can be had if it were not a drawing of space itself, but
spacetime. Time can be assumed the radial dimension, and space in each moment is a spherical surface (it is a 2d analogy, of course)... all moments together forming a spherical volume... but it's still not the best representation, as the curvature of space may not be so directly related to time...
I doubt this is what you meant in the first place, and I don't see how it would relate to the second image, so I'll put a stop here.
Gah.. my head hurts now. But I may come up with something else in a few days...