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?