I think there are a few difficulties with doing as you suggest. One is that the sections already flow under a H2 (Accesshide-topic outline)
All sections are LIs under that H2. I don't know what purpose that H2 serves, however, so maybe we can do without it and shift everything else up.
The other problem is that I think there's a lot of CSS and Javascript that hangs off those H3.sectionname tags not just in core Moodle but in plugins also. It's a potential breaking point that may have quite wide repercussions.
Can't really add anything about the Adaptable theme issues except to say that you can override the core renderers in a theme so you may be able to change the HTML structure in your theme (assuming you make a child theme of adaptable). But would will still make the same point about unintendingly breaking other things.