Here is the fix:
If you unload the material library from the template, save it, close Revit, reopen, create a project from your template and add the Material Library there: not a problem. It’s all there.
Now getting back to answer number one: it stays there. The above workflow is a definitive solution to the problem. Once loaded into the project you can reload the Material Library in your template. Then and when you create a new project from that template there won’t be a problem. It’s like the entire problem never existed.