mail you send via SMTP goes into your INBOX
Okay, that's weird. I see where they're coming from, sorta, but that's awkward.
The mail client can't delete mail from the POP server
Okay, the POP client can't delete mail out of the Gmail inbox. I'll give you that. But that isn't a violation of POP. It deletes it out of the POP box, though. Effectively, the web interface and the IMAP interface are synchronized, and the POP interface is really just a copy of everything that's sent to the "real" account. The POP RFC explicitly states that it has no interest in what goes on behind the scenes.