I agree, the suggestion that anyone could/would be able to just throw out everything they know about arithmetic is sketchy at best, and even it's sketchier to make it seem as though a result like Euler's formula could purely be derived from geometric intuition. Sure, it "makes sense" in the video, but I don't think it's possible to truly grok and appreciate the result without having done any calculus. The real joy of math isn't in knowing, it's in proving.

Anyway: as for proof methods, I like the Maclaurin series trick with this one. Too bad the internet hasn't come up with a universal/nice way of writing math notation in a readable format yet, I'd like to sketch that proof here...