Facebook – Is a Reddit-like self-maintenance possible for a Facebook group

Facebook Help Center says:

Our system automatically deletes groups once they're empty (no members). If you created the group, you can delete the group by removing all members and then yourself. Note that if you were not the first admin and creator of this group, you will not be able to delete the group unless the group creator has left it voluntarily.

You must have a new-style group. Only old-style open groups (which cannot be created any more) can be joined without approval.

Previously, open Groups were similar to Pages and anyone could join. They were so similar that a lot of people were confused as to which they should use, and they would often complain because they created the wrong kind and would want to switch (which is not possible). Facebook redesigned Groups in October 2010 to make them better for small groups of friends, with features such as group chat and new notification options. Any member can add new members from their own friends list, or others can request to join with group admin approval. Pages on the other hand are designed for large numbers of people, and for any case where you want anyone to be able to join without approval.

Although each group member can change their own settings, by default all group members are notified when there is any new content from any member in one of their new-style groups. This is very helpful for small groups of friends, however it does not really work if anyone can join. The reason is that spammers use scripts to create Facebook accounts and join old-style open groups and become fans of pages in order to post spam on the group or page wall. At least with a post on a Page, most people will not see it unless they explicitly visit the page, and hopefully a page admin will remove the spam and possibly ban or report the spammer by then. Also a page admin can set posts from others to be hidden by default, if they don't want to clean up spam. And pages and old-style groups do not have a group chat feature to worry about protecting from spammers.