This site offers a public, non rate-limted API. IANAL but I'm reasonably certain that's authorization for anyone to use the data as long as they do so through the API. It certainly isn't the case that you need explicit legal permission to use Hacker News comment data in your project.

There have been tons of alternative frontends and projects using HN data over the years, posted to Show HN without an issue. I think their primary concern is interfering with the YCombinator brand itself. "the site" and "site content" referring to YCombinator and not HN specifically.