Also, it's patently absurd someone would include Fitch's Paradox, a piece of philosophy, on a list of "counterintuitive facts."
First, we don't know that spacetime is quantized; that's a plausible speculation but we have no theory of quantum gravity.
Second, "quantized" is not the same as "discrete". A free particle in quantum theory is "quantized" but the spectrum of all of its observables is continuous.
When comparing something of a certain dimension with something of a higher dimension, it's not at all surprising that the lower one can be infinite and the higher one finite.
Usually it's phrased as "a finite amount of paint can paint an infinite area." But why do I need the Horn to realize this? It works in the Horn only if there is no lower limit to the thickness of paint. If you accept that, then I can take a drop and paint an infinite plane with it. Why do I need the Horn to demonstrate this?
So if you think of it as a bucket that can't hold enough paint to cover itself, that is at least a little surprising.
If you don't allow for infinitely thin paint, then no - Gabriel's Horn surface cannot be painted even with an infinite amount of paint.
I believe the point of the exercise is an introduction to curves with infinite length but finite area under them, in order to expand one's intuition about such objects, which is then transferable to other examples like space-filling curves.