Ahhhhhh HA! Look at this condition:
"Triggers are based on the date and time an email was received. Moving an email to another folder does not change the received date property value, so the triggers will skip any email that was received prior to the latest successful run."
So...hm...yes, I see the issue. If I drag-and-drop all my e-mails in chronological order (or, immediately, as they appear in my inbox), this flow should work. If I start at the top of my inbox and drag e-mails to the folder, the flow will (probably) miss all of the e-mails except one, because it will look at all the other e-mails and say "The date of my most recent successful run is later than the date of that e-mail, so I do nothing.
I can think of a few ways to work around the issue - some of which you described - but the best thing to do might be to just set up a rule that auto-drops the incoming e-mails into the target folder. I'm not familiar with the ins and outs of the "Get Emails" action but I could foresee that being a potential solution as well.
I'd like to wait a couple days before I mark this as the accepted solution, just to confirm that this is indeed what's happening, but this is a great answer and it seems highly probable that this is the issue.
Thank you!