Hi Cody,
actually I am not able to figure it out and have now spent way too much time on it.
I am testing the update on two devices. Both have TeamViewer 15 installed but are having different minor versions. On one of the devices I have tried multiple minor versions to see if it makes any change. It doesn't.
So right now I am stuck with an installed version (15.6.7) but the reports (e.q. Compliance

are reporting the update as "not required" and therefore wouldn't show up. I quadruple checked the applicability rules / registry.
There's a Key (HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\Microsoft\Windows\CurrentVersion\Uninstall\{36A33BB9-E44B-45EA-A4FD-E861A103FF97}) with DisplayName set to "TeamViewer" and DisplayVersion to "15.6.7.0".
Both clients just work / report fine to any SCCM component, no obvious issue present in log files related to software updates.
As I have tested my application deployment of the new version on both of those clients before the testing of the update it may have corrupted something.
Therefore I went and tried another client which had an even older version installed (15.1.3937.0). The update was available to install yet the registry key never changed to anything newer than 15.1.3937.0. However the local TeamViewer.exe showed 15.8.3 in the files' details tab. Checking the client with the "Client Center for Configuration Manager" (3rd Party Tool by Rob Zander) there is no TeamViewer 15.8.3 Update marked as applicable or installed. But there's an update for TeamViewer 12(!) marked as installed.
At the moment I think there is something really weird happening on our side and it has nothing to do with your published updates.
My last idea until I'll just deploy the latest version as an application:
We are using the msi installer with options for management assignments (API token etc.) - are there any known problems with patching the MSI versions with the updates provided by you? I would have thought it was working, but obviously things are bit different.
Best regards
Fabian