I was about to make a request on your behalf, so I did a topic search and found many similar requests. The response in general was that it's theme related and not the concern of core developers. The impression was given that comments on pages was easy to implement.
Aware of your issues, I created an author page as you did on my test installation. To my surprise, I found that with my theme, a heavily tweaked twentyeleven, comment support is already in place. It works properly with out my doing anything!
Now that I think about it, when you request a page, a query is generated that will return that one page. Thus comments should be assigned to that page's ID automatically, as that is the current "post".
There must be something odd about your theme's comment form that is preventing it from finding the proper post ID. So hopefully, a solution will be easier than I thought initially after your first misstep.
I did notice my page template uses
<?php comments_template( '', true ); ?> instead of get_template_part(). You should try that and see if it helps.