The reason that the proof isn't geometric, is that the algebriac proof is a proof that Euclidean geometry is incomplete. How can you use a language (any given language!) to express the idea that the selfsame language is incapable of expressing a certain concept?

You can draw a picture of trisecting an angle using an ruler (with cube-root markings) or an Archimedian sprial, which are clearly more powerful than purely Euclidean geometery, but how can you draw a picture of it being impossible without something like this?

How do you draw a picture of something that doesn't exist?

You can draw pictures of what does exist, like the symmetries in Arnold's proof of unsolvability of the quintic https://mcl.math.uic.edu/mcl.math.uic.edu/wp-content/uploads... and show that those symmetries can do things that radicals can't.

I don't know of a similar visual for non-trisectability of angles.