They don’t own it, the individual posters own the content of their own posts, however, from the reddit terms of service:

When Your Content is created with or submitted to the Services, you grant us a worldwide, royalty-free, perpetual, irrevocable, non-exclusive, transferable, and sublicensable license to use, copy, modify, adapt, prepare derivative works of, distribute, store, perform, and display Your Content and any name, username, voice, or likeness provided in connection with Your Content in all media formats and channels now known or later developed anywhere in the world. This license includes the right for us to make Your Content available for syndication, broadcast, distribution, or publication by other companies, organizations, or individuals who partner with Reddit.

And with each of those rights granted, Reddit’s lawyers can defend those rights. So no, they don’t own it “just because they ran the servers” - they own specific rights to copy granted to them by each poster.

(I don’t like this arrangement, but ignorance of the terms of service isn’t going to help someone who uploaded a full copy of the works they have extensive rights to) On this subject I think there needs to be an extensive overhaul to narrow what terms you can extend to the general public. The problem is I straight up don’t trust anyone currently in power to make such a change to have our interests in mind.

It would be inviting a lawsuit for sure. I like the essence of the idea, but it’s probably more trouble than it’s worth for all but the most fanatic.