I believe this site is missing a very important thing, direct links to the different categories with a list of papers.
This is at least how I (and I believe many others) browse arXiv. I open it up in the morning, scroll through a few categories and open a few papers that look interesting to me.
I could see myself using alphaxiv for that, and then, if there's a comment section, I might even read it, and, who knows, leave a comment. But there's no way I'm going to be changing the address or going to some other site to search for papers just to see whether there are some comments.
ps: I see the extension adds a "discussion" link to arxiv, it is a pity that it is only available for Chrome.
It sounds like what you want is scirate. As far as I understand from this post this new thing is just scirate but lacking the interface you're talking about here.
Kinda related, Hypothesis (and Diigo iirc in past) has an extension/bookmarklet that can provide an annotation/comment overlay on any web page/pdf. Guess what is needed for arXiv discussion is this overlay but smarter, that is knows a paper pdf and web view are the same, and abstract page is connected to them.
I could see myself using alphaxiv for that, and then, if there's a comment section, I might even read it, and, who knows, leave a comment. But there's no way I'm going to be changing the address or going to some other site to search for papers just to see whether there are some comments.
ps: I see the extension adds a "discussion" link to arxiv, it is a pity that it is only available for Chrome.