Even if you say you don't need to store emails, there is still a chance of leaks via logging systems or simply programmer errors, or some caching layer. It's likely your security protocols aren't as robust as Google's. (You mention "https" as the primary reason this is secure. Much wow.)
Given the importance of email there is no way I could trust another closed system with my email. I'm sorry if that sounds harsh, I also want to see alternative email frontends because gmail isn't cutting it.