Let me take on a different meaning of "introducing" and argue why I think Euler's identity is a great way to introduce the concept:
I teach primarily remedial students in grade 8, 9, and 10, and I put the following classroom poster:
$ \sqrt-1 \space 2^3 \space \sum \pi...$ and it was delicious!
None of my students need to know about $i$ or $\Sigma$ (yet) -- and a few of them are disappointingly clueless about $2^3$ and $\pi$ -- but nearly all of them were interested in the poster and wanted to know what it meant, and some students even did a bit of research outside of class.
Now, I have not introduced these concepts in a mathematically meaningful way. My students don't know what half of the symbols mean. But by casually mentioning to them that $i$ is not on but above the number line and $\Sigma$ is like + but not really, I hope I can pique their curiosity.
In the same way, I think that using Euler's identity to dive into a long lecture might undermine students' enjoyment, but perhaps leaving the identity for students to think about as they progress through learning about $e$, $\pi$, and $i$ would be motivating or at least interesting.
Finally, it must be noted that Euler's identity really is much better than the lame math pun I put up in my classroom because it's true! That fact might be lost on students, but it shouldn't!