Gmail shows archived (previously-snoozed) messages in the Inbox
If you snooze messages, wait for them to return to the Inbox, and then you archive them from any email client, such as Mimestream (Gmail API), Apple Mail (IMAP), etc., then there is a Gmail server-side bug where those messages continue to appear in the Inbox.
This bug has been reported to Google, and we have escalated it through contacts we know at the company. To express interest in seeing this fixed, please “Star” the following public Gmail bug (though adding comments is discouraged): https://issuetracker.google.com/issues/151714665.
Confirming that you’re seeing this bug
The easiest way to confirm if you are seeing this is to conduct a search for “in:inbox” in Gmail (or “in:inbox category:primary” if you use Categorized Inboxes). If search doesn’t return the messages that seem stuck in your Inbox, then Gmail itself is showing an inconsistency, and you have likely hit this bug.
The only workaround is to just go ahead and archive them through the Gmail web UI.
Resetting the Mimestream cache will not help, because the Gmail API already returns those messages without the Inbox label.