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.)