Update: I believe I found the solution.
When I originally opened HBO Max in StreamFab, I had selected HBO Europe from the list of VIP services. With that option, the video would play fine inside the StreamFab browser, but the Analyze button never appeared at all.
I just tried again, but this time I instead selected the main HBO Max service and the button appeared immediately.
So even though my HBO Max account is European, I still need to log in through the standard HBO Max option in StreamFab. My guess is that HBO Europe may refer to the older regional platform rather than the current HBO Max setup.
Just posting this as a reply to my own post in case anyone else runs into the same problem in the future.