Settings API is (also) about the modularity of the settings, settings can be removed (never heard of anyone do it but it is possible) or amended (my favorite technique right now).
When you separate the HTML and PHP you are breaking the modularity so maybe the setting API should just not be used by you.
As with many other things you can probably hack it to work with using all kinds of filters, but the question is why to use it at all if it is against your preferred development practice.