What artefacts of the ant on a wire analogy for string theory's curled extra dimensions are due to the analogy breaking down?

I'm trying to understand the ant on a telephone-wire analogy that's used to explain why we might not see the extra dimensions from string theory. I'm quite comfortable with Newtonian mechanics in 3D, and have read Brian Greene's Elegant Universe.

So, here's what I don't understand. The analogy seems to make a lot of the idea that from a distance, an ant tracing a helix on a wire looks like it's just travelling along a wire. I.e. a 1D path. But, if we zoom in, then we see that it's tracing a helix. Why does the zooming matter? Isn't a helix still just a 1D (1 parameter) curve no matter whether it's embedded in 3D or beyond?

An ant is a 3D object (as far as we know) and in going around a wire, it's tracing a helical path in 3D. It's never leaving our usual 3 spatial dimensions. It's never travelling perpendicular to all three of those dimensions at once.

Ok, so what about a point? If it ever moves in a direction perpendicular to our 3 spatial axes, then it disappears, since it's left our 3D "slice" of the higher-dimensional space.

Ok, so let's consider something higher-dimensional than a point. It may look like a point to us if its intersection with our 3D slice of the hyper-space is a point. For it to appear to be moving along a curve in our 3D space, it would have to move in the higher-dimensional space such that its projection into 3D was always a point at any instant, and also so that the 3d location of this intersection point moves continuously along the 3d path we observe in our 3D slice.

Now, where does the "curled" or "hidden" nature of these extra dimensions come in? Are we just saying that for all the configurations allowable for these higher-dimensional objects, a lot of them lead to intersections with our 3d space that look like movement rather than say radical shape changes of the intersection with 3D? I.e. we'd easily notice there were extra dimensions if these objects were drastically changing shape in 3D like the standard example of a 4D cube projected into 3D. But if, instead, most of the time their higher-D configuration changes project down to linear motion / rotation in 3D, then we'd have a harder time realizing that we're looking at higher-D objects moving in complicated higher-D trajectories. We'd just assume they're 3D objects flying around in 3D.

Is there something essential I've missed? Have I introduced anything unnecessary?

Thanks!

-Look from far away $\rightarrow$ low energy limit
-Zooming $\rightarrow$ going to high energy experiments
-1-d wire $\rightarrow$ 3-d extended space
-2 small directions on the wire $\rightarrow$ 6-d compactified space (Calaby-Yau)