Why do sites that host papers not have a comments section for each paper?

There are a few third-party sites where you can do this. For instance, https://paperhive.org/ and https://scirate.com/. None of them has a large user base, at the moment. There used to be more (http://www.arxaliv.org/, https://selectedpapers.net/) but they seem to be down.

(Disclaimer: I am a friend and a former colleague of one of the founders of Paperhive.)

Reason 1: Many publishers (still) see themselves simply as content providers. A reader goes to the publisher website, downloads a pdf and leaves. A publication is understood as a static (dead?) document, so post-publication activities are not encouraged technically.

Reason 2: The publisher website landscape is very fragmented. Even smaller scientific communities have to go to multiple publisher websites to get their content. So, a researcher would need multiple logins to add comments on the different websites. A high barrier even today.

Reason 3: A simple forum-like comment covers only a few of the post-publication discussion use cases. It is good for general opinions and corrections. Annotations, however, are much more targeted and also cover use cases like corrections of specific mistakes in formulas, paragraph-level questions and statements, discussions around data and images, proofreading, and referencing of specific parts of a document (example of direct linking at PaperHive: https://paperhive.org/documents/jKRG3MqMuj66/revisions/QOdeW6lxZMHk?a=s:Vjw91UM7br1A).

These are some of the reasons why we at PaperHive [Disclaimer: I'm one of PaperHive's founders] chose to build a system that is connected to publishers' websites but enables the communication on a separate cross-publisher platform entirely dedicated to reading and discussing. We have a small widget that informs readers on a publisher website of existing discussions (see example on a partner's website: https://www.growkudos.com/publications/10.1016%252Fj.neurobiolaging.2014.04.026) but the actual reading & annotation happens on PaperHive.

All public discussions are under the CC BY license and we follow the W3C open annotation standard (which is the foundation for interoperability and portability). We have an API for the export of annotations (documentation in progress) and have on our roadmap to archive the discussions at one of the two most-widely used systems in publishing (Portico or Clockss).

There are a couple issues with this that I've both seen and experienced:

• It's probably not worth it, as few papers actually draw comments. Pubmed Commons, for example, means you can leave comments about a paper on it's Pubmed entry. I looked up some of my papers...there isn't a single comment. Not one. And while I'm not the most prominent researcher in the world, we are talking about a body of papers with 500+ citations. So...why bother with implementing this?
• If you do get comments, you're vulnerable to things. For example, the anti-vaccination community launching a spam campaign. While you can restrict the comments to subscribers, what about open access journals? Crackpots with library memberships? Many of the same arguments that you should be able to comment apply to the public being able to comment. And that means moderation. Whose going to do that?
• What are the obligations to commenters? Am I free to ignore them? Or, if J. Random asserts there's an error in my paper, do I have to address it. How much do I have to address it?

TL;DR: There's significant costs and not a lot of clear benefit, especially when there are ways to contact an author directly.