I have noticed issues where the gmail priority inbox is used. Things go to folders like family and friends, important etc and never appear to make it to the inbox when POP comes calling.

Messages that Thunderbird POP doesn't see are in the regular Inbox on the Google webmail. And, K9 mail POP on my Android phone sees them and downloads them. Must be something unique to Thunderbird.

re :And, K9 mail POP on my Android phone sees them and downloads them. Did K9 mail POP on Android phone access and download that email before Thunderbird ?

Gmail can be weird when it comes to POP. Gmail removes the messages from the POP download queue after they are downloaded, so they cannot normally be downloaded again. Maybe this explains what is going on. read info under 'I want to download emails on multiple email clients ' at the following link: https://support.google.com/mail/answer/7104828?hl=en&visit_id=636894836272661741-2868050981&rd=1

Or maybe you have an Anti-Virus product on computer that is scanning incoming mail and is having an issue with emails from a particular email address. Try switching off email scanning and then doing a 'Get Messages' to see if it downloads.

Yes GMail appears to be weird when it comes to POP. A couple of days experimentation says that GMail appears to be removing "some" but not "all" new messages from the download queue (when I poll the GMail POP server from the first device I sign in to today). Then when I poll it from another device poof, they are gone. The "recent" mode sounds intriguing; maybe it can be made to work.