Hi,
That is a very good question!
Originally when this plugin was created, adding support for the template editor to any theme was to allow users to see and play around with the template editor more of a matter of curiosity because at the time, no FSE themes were available.
Now that “Twenty Twenty-Two” and other FSE themes are available, this plugin’s purpose in life became apparent to enable users to be able to manage the templates in those themes as currently, this facility is “tricky” at best (although I’m certain that work to make this easier is ongoing in WordPress core).
So the answer to your question (especially without knowing what theme you are using) is that if you are using a non-FSE theme with this plugin expecting it to FSE-enable your theme automatically, then I’m afraid that’s almost certainly not going to work and it’s not surprising that things don’t work as you’d expect.
Having said that … ?? … we are currently working on an update that will allow you to manage the template parts (not just the templates) and so it may be that once you are able to add template parts (like “footer.html” or “header.html”) that this does what you require!
I’ll let you know when we’ve pushed the update.
Oliver