Excellent post! It reminds me of this (slightly paraphrased) line of Keats: "axioms are not axioms until they are proved upon our pulses."
On the practical impossibility of reasoning from first principles, here's a lesser-known but much more modern than Aristotle example of this phenomenon, which shows that this is very much still a problem in science.
In 1995 David Auckley and John Cleveland axiomatized origami and proceeded to prove that the set of points in the plane constructible via origami is a proper subset of the points constructible via straightedge and compass.
However, they weren't origami experts and their axioms missed a number of real-life possible folds. More realistic axioms were proposed by various other mathematicians, now known as the Huzita-Hatori axioms. Using these, which reflect actual real-life folding practice, it turns out that the set of points constructible via origami is actually strictly larger than the straightedge-compass constructible points. In particular, a whole new family of polygons becomes constructible.
References:
[1] Auckley and Cleveland: https://arxiv.org/abs/math/0407174
[2] Huzita-Hatori axioms: https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms
