Do Peano curves provide a counterargument to Grothendieck's critique? This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be regarded as exhibiting the naturality of phenomena like space-filling curves previously regarded as odd?
To clarify, the notion of naturality is not mine but rather is implied by Grothendieck's critique of the foundations of topology; see the earlier question linked above (perhaps he even uses the term natural, I would have to double-check). According to that view, odd phenomena like space-fillling curves are an artifice of the foundations. The question is whether the Cannon-Thurston paper finds them to be actually useful for something, which may be an argument against Grotendieck's criticism.
 A: Here I am expanding Doug Zare's comment into an answer.

The original poster asks:

Can the paper Group Invariant Peano Curves by Cannon and Thurston be regarded as exhibiting the naturality of phenomena like space-filling curves previously regarded as odd?

and

The question is whether the Cannon-Thurston paper finds them to be actually useful for something, which may be an argument against Grothendieck's criticism.

I do not have to hand Grothendieck's definition of "natural" nor the poster's definition of "useful for something", so I will substitute my own.

I take "naturality" to mean "without making many choices" and thus "having unexpectedly good properties".  For the various Peano curves, many choices are needed.  There are more-or-less simple recursive constructions, but this only underlines the fact that you could bolt these together in various ways to get uncountably many examples.
On the other hand, to obtain a Cannon-Thurston map, only two choices are required: first a fibered hyperbolic three-manifold $M$ and second a fibering of $M$.
With the real choices out of the way, the construction of the CT map is as follows.  We choose (but this does effect the eventual answer) an elevation of $F$ - that is, an embedding of the universal cover of $F$ into the universal cover of $M$.  The former is a copy of $\mathbb{H}^2$ while the latter is a copy of $\mathbb{H}^3$.  Any hyperbolic space $\mathbb{H}^n$ has a boundary at infinity - this is a copy of the sphere $S^{n-1}$.  Thus we obtain a map $\phi \colon S^1 \to S^2$. That this map is continuous requires work.  That it is sphere-filling (and finite-to-one, and has certain invariance properties) is easier.
Note that if $N \to M$ is a finite cover, then the fibering of $M$ induces one of $N$.  Let $G$ be the fiber in $N$. Thus there is a covering map $G \to F$.  The CT maps associated to $(M, F)$ and $(N, G)$ are then isomorphic, and there is a diagram expressing the commutivity.

This is also useful for various things.  For example, Mahan Mj has used the existence of (more general) CT maps to prove that the limit set of a (finitely generated) Kleinian group is locally connected; see his talk at the 2018 ICM.
