The Maclaurin trick is what I'd call the "standard proof", and I think it's the source of a lot of the lack of intuition surrounding e^i... It's exactly what you call it, a "trick", where you prove two rabbits are identical by transforming them both into an infinite number of hats.
On the one hand, it's all sound logic and algebra, but on the other hand, I certainly don't blame Randall Munroe for seeing that proof in high school and going on to say "I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave."[0]