Additional features, settings, or issues not covered by specific Microsoft Teams categories
The following is the answer I got back from Microsoft:
The Website tab is not a supported mechanism for integrating content into Teams. The Website tab has always been a best effort mechanism and we cannot guarantee that the content embedded in it will work as expected. If the content doesn't work, then customers should click on the "Open in browser" button in the warning header to launch the pinned URL in their default browser where it should work as expected: From a technical perspective, the Electron webview that we used to embed websites into the Classic Teams Client has been deprecated and we have switched to using a vanilla iframe in the New Teams Client. That means a number of sites that worked in the Classic Teams Client will no longer work in the New Teams Client. Customers should use dedicated apps from the Store to embed the content where available or load the sites in their browser if no app is available.
If this is true; this would take the only good thing about the teams environment away. I am using it to show document libraries are a certain level (as in a few folders deep rather than at the root folder), and no app will achieve this unless I am missing something!