Pythagorean Theorem intuition

I actually have a degree in Pure Mathematics, and this has always bugged me. Like a tiny, hardly-noticeable stone in your shoe you just can't get out.

Despite being at the core of geometry, arithmetic, and infinity, despite being able to prove it a dozen different ways, and despite it being the first "formula" anyone learns and one of the oldest known mathematical ideas - I can only see shadows and silhouettes of its truth.

Looked at from the point of view of geometry, I can see the statement plainly as a statement regarding the intrinsic notions of angle and area (using the proof of similar right-angle triangles) - however, the last step requires a correspondence between the notions of length and area to make complete (namely, the area of a shape increases quadratically as its scale increases linearly). This has always bothered me, since, the Theorem is an idea solely between angle and length.

There are, of course, simple proofs that don't depend on the notion of area (ex: Pythagorean Theorem Proof Without Words 6), however, sadly, they don't further my intuition, despite being clever/cute.

Staring at the statement blank in the face, I understand it far less intuitively then I do half of the ideas in algebra, analysis, number theory, etc.