I've been asked that several times, even by my fellow moderators.
There are several reasons, but it mostly boils down to "Interaction".
LiveJournal offers a level of discussion rarely seen on other sites. Forums often have rather hard-to-follow comment threads (if they have threading of comments at all). Facebook truncates large comments, and formatting a post with the proper evidence would be next to impossible. Tumblr's comment system is rather...disorganized. Twitter has a character limit, though we're working on setting up a feed for notifying when a post is made. Art sites don't allow call-outs, which is what our 'bewares' are often called.
Finally, using a our own site would require users to create a login for yet another site. We feel that would discourage the level of discussion we have here on LiveJournal. Here, at least, you can log in with your Facebook, Twitter, and Google+ account, thanks to Livejournal's connect widgets. No LiveJournal account needed.
Because Artists_Beware isn't just about tossing up a post about a bad experience with an artist, commissioner, or business. We offer advice and suggestions on how to resolve these issues, sometimes even debunk the issue when other proof shows up due to members' questioning. That wouldn't happen with a "post-and-walk-away" site.
There are other reasons we stay on LiveJournal, and yes, there are also things we don't like about LiveJournal, but not enough to drive us away (yet).