Got it! So if I understand correctly, the Discourse Patreon plugin is not currently able to add free Patreon members to Discourse groups.
Patreon does allow me to list free members as a labeled column in its audience manager and I can export a CSV showing this as a tier too:
I don’t know what kind of data the Patreon API returns on calls against it but I do wonder if this is a solvable problem. But anyways that veers into bug report/feature request territory and my original question has a clear answer now, so I’m going to mark this topic solved and change the title to something easier to search for!
Edit: Ah shucks, I don’t think I’m a high enough trust level to edit my title right now. I was going to go with Patreon plugin is not adding free patreon members to group if the fancy strikes anyone to do that 
Thanks for the help, @jessii !