@JeremyHellstrom,Thanks for posting in Q&A.
From your description, I know that you want to replace "Teams Classic" with the new Teams on your taskbar using Intune but get some problems.
To clarify this issue, here are some information you can refer.
1.Please be sure that the new Team has been installed successfully on targeted device.
2.When the app successfully installed on targeted device, you should pin it to the start menu manually first, then you can run command Export-StartLayout -Path "C:\software\GetIDorPath.xml", the new Teams will show up in the xml file.
3.To pin app on the taskbar, for Classic Windows applications, use shortcut (.lnk) files. And for Universal Windows apps, use the Universal Windows app user model ID.
Here are some links about how to customize Taskbar and find application type you can refer.
https://learn.microsoft.com/en-us/windows-hardware/customize/desktop/customize-the-taskbar --Windows 10
https://learn.microsoft.com/en-us/windows-hardware/customize/desktop/customize-the-windows-11-taskbar --Windows11
Non-official, just for reference.
Hope above can be helpful. If there is any update, feel free to let me know.
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