His student's "proof" is an illustration, not a proof. There's no way to know if its circular or simply unfounded, since it is purely an appeal to intuition. That's only a part of mathematics. The correct thing to do, mathematically, is to validate the intuition by formatting it as a proper proof, based on non-circular axioms and theorems. In the paper itself he admits that his student's work was incoherent and needed him to rewrite it.
He's right that 2 column geometry proofs are ugly, and could be presented better. This has been known for centuries.
https://www.c82.net/euclid/en/book3/#prop31
For children and for starting out, the pictures are great. But for mathematics, pictures are extremely limiting. 2D and 3D are notgod models for N-Dimemsions. ("Spiky balls" , for example).
Mathematics is far, far more powerful than human eyes. The amazing thing about geometry is that the whole thing works without pictures! A blind person can be a great geometer, because geometry is axiomatizable. Meanwhile, Euclid's Elements, while an incredible achievement in its day, is not well-founded, relying on unstated axioms.