Editing index.html will not work because Joomla doesn't know anything about the file, it's manually parsed via the JoomlaComment component (it actually doesn't even load the HTML as a Web Page, just as raw text).
EDIT: Wait, perhaps the article tag should work in the JC template because it's executed onPrepareContent... I'm not sure. But you will need to move the {loadposition relitems} tag to
underneath the
{body} tag in the JoomlaComment template - it already says in the comment template that "only data within the {body}{/body} tags are parsed by Joomla" (or something like that) and everything above it is for
Template Preview only.
I have no idea how to re-code JoomlaComment into a module format (I've never written a component before) and Daniel already has a great deal of work... but I am interested in this. Thankfully, there are already some Component > Module wrapper solutions for you to try to see if it is at least possible (and if we're lucky it might work as a module straight away) but if you don't mind testing then try out one of the many extensions on the
Embed & Include section
of Joomla Extensions Directory - they have extensions there that can help, such as
Plugin In Module
which could allow the JoomlaComment plugin to execute in a module position instead of within the main content. Or perhaps the
Component as Content
extension might work better? Those two stood out to my eye, I might just test out the Component as Content one in a few days if I get around to it.
If you can help us with this, please do so - try a few of those extensions on a test site that might sound like a solution and tell us which one works the best for your Joomla site and if there are any bugs from using the one you decide on. If you do this I will try to bugfix it, and might even code up a new "Module Wrapper" option based on the mostly-working, generic wrapper extension you find to include in newer JoomlaComment releases