I’m forwarding to or retrieving email via POP3 at Gmail, and when I send a test email from the same address that will ultimately be receiving it, it doesn’t work.

If you’re forwarding to Gmail or importing via POP3 to Gmail, you can’t send a test email from the same Gmail account that the email will end up at and expect it to pop up as a new email in your inbox. Gmail doesn’t work that way.

If you have an email at a Gmail account and you need it to end up at that same Gmail account, and you send it from that Gmail account, you already have a copy of it. So then why does it need to be at the top of your inbox? It doesn’t because you’ve already read it (because you sent it), so the test isn’t a real world use case.