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I was wondering why my {exp:comment:entries weblog} tag doesn’t display my comment entries on the index.php main page? Is it not possible to nest this inside the {weblog entries} tag?

Comments are associated with a single entry, your index is more than likely a multi-entry page. The commenting system keys off the last segment of the URL, the entry ID or entry title. That’s obviously not part of the URL on a multi-entry page.
There is a Wiki article for a method to show comments on the index or multi-entry page if you want/need that. It comes with a warning though… Wiki

Also, you are correct that you should not be nesting your comments entries tag inside the weblog entries tag.

From the docs.

Please note that the comment entries tag should not typically be nested inside of a standard {exp:weblog:entries} tag.

I am not sure why you need to show comments on the main page (or even exactly what you mean by main page) but typically the reason for this would be to show the most recent comments in the sidebar.  To do this you would use the dynamic=“off” parameter in conjunction with others in that list.