Okay, fairy nuff.
In that case, I would probably start with writing an SMTP or IMAP proxy first. It will teach you everything you need to know about the protocols, and you can reverse engineer the protocols using a client that already works.
It would give you a tangible project outline, which I believe is often critical to not lose motivation or interest.
If you accept using libraries, there’s the imap crate, the mail_send crate, and samotoo crate that are worth looking at.
I think you’re misguided about the APIs. Gmail supports IMAP and SMTP. Proton supports those too if you run an encryption bridge on your computer. Fastmail supports IMAP/JMAP/SMTP (they invented JMAP to try and innovate).
Email providers most likely must provide SMTP and IMAP due to compatibility requirements with Apple Mail and other clients.
Email is ridiculously complex—the technology is dead simple, but the number of exceptions and (undocumented) rules you need to abide by or risk getting banned by half the internet without being told is nothing to sneeze at.
I should know: I have built multiple support platforms that worked through email (amongst other channels).
You mention wanting to start at the SMTP level, and then building a Qt interface. So you’re going to write an SMTP client, an IMAP/POP3/JMAP client, a storage engine, a user interface, and a better search system, all on your own? You’re describing a gargantuan task.
No offense, but each one of those could be a project on its own. You probably think they’re all simple tasks (they’re not), and that you can follow a few RFCs to get things going (you can’t), and that it’ll be easy to debug (it won’t). Finally, I think you’re underestimating how large people’s email maps get.
Why not write a plugin for Thunderbird that improves the search?
The irony is that Reddit thinks their internal search can in any way, shape or form compete with Google or other actual search engines. People use Google to find stuff on Reddit because Reddit's own search is hopeless.
First sentence on the first hit when searching for “Gmail smtp imap”:
https://developers.google.com/gmail/imap/imap-smtp
What you’re referring to is the fact that GMail has apparently disabled authentication using username + password for SMTP/IMAP. I would assume that application passwords still work fine as a workaround, even if they don’t mention it specifically.