You're neglecting the tools and bots that use the API, which are heavily utilized by most mod teams. One of the pillars of reddit is unpaid moderators, and if the tools that make that job doable on the scale of reddit stop working, then you will see a mass exodus of those unpaid moderators. That means the death of most of the big, well moderated communities like AskHistorians, AskScience, AITA, etc.
I've already seen many of these subs having moderator led discussions about relocation options for the communities.
Reddit has expectations of what moderators are to do, and has expectations of what they are not to do, and will remove them from roles if they fail to meet those expectations. That set of expectations would make them employees if compensated.
As for liability, the Ninth Circuit in Mavrix v LiveJournal held that if an agent of a user-content-hosting ISP (social media) has the means and opportunity to moderate, they also have the means and opportunity to interdict reasonably known copyright violations, and failure to act on those would jeopardise their DMCA Safe Harbour.
And there’s a lot of registered copyright holders that will 100% line up to be a creditor on statutory damages.
Reddit moderators do not directly deal with DMCA takedown requests. If Reddit is presented with one, they will take the offending post down directly. Moderators can be suspended or removed, however, if they encourage rule-breaking behaviour in a subreddit (such as by soliciting content that results in DMCA takedowns).
The primary social role of moderators is to curate the community. That involves enforcing some site-wide rules, but it also involves more local rules like "stay on-topic." It wouldn't do for a forum about NFL football to be taken over by discussion for The Bachelor, even though that's not actionable at a site-wide level.
I've already seen many of these subs having moderator led discussions about relocation options for the communities.