Your first paragraph is already implemented in something called SPF. It already works using the existing DNS infrastructure. The problem is that creating SPF records is effectively voluntary, so operators of mail servers are only able to use existence of the records as a way to increase trust, and not using the absence of the records as a way to decrease trust. Until everybody is on board with it, unfortunately, it's usefulness will be limited.
And, just for clarity, a POP3 "server" doesn't accept mail. POP3 is a protocol for retrieving mail from a mail server that likely received the mail from another mail server via SMTP. SMTP is the problem, not POP3.
And no, it won't solve the NSA problem, or the Google problem. They'll just build bigger and faster computers to decrypt the emails.