Its unlikely that we'll lose the site character of uploading others work. Danbooru relies on one person endorsing the art of another by bringing it here. Self-uploading is few and far between enough that it tends to be done by those who feel they have to prove their art is good enough, which means likely it isn't. That isn't enough I think for an outright ban, but consider this:
A great artist never has to worry if their work is featured here. Contributors will upload their work within moments of posting to Pixiv/Deviantart/Tumblr/Seiga. OK art gets posted by member level users who find younger artists and people who tend to get missed or aren't followed by the contributors for the lack of mainstream content. Weaker artists are usually uploaded by people who don't have experience in choosing content. Chances are a bad artist won't know what to look for in a great art piece.
An artist self-uploading is in a majority of cases, an artist who doesn't feel that anyone likes their work enough to put it anywhere. They tend to miss that what matters most is to focus on the art itself, and let someone who can see its merit come and post it for them. They're not malicious for wanting to post it, but they are naive. We should encourage artists to just focus on their work and not worry about putting stuff here. If someone likes it enough, it'll get uploaded anyway.
I think we should emphasize in the wording that "If you're an artist who wishes to have their work featured on Danbooru, please focus on refining your work and uploading on Tumblr, Pixiv, Seiga or Deviantart, and let our users decide which of your works will be featured. Danbooru prides itself on featuring the best Japanese art from across the web, and a good artist can expect not to have to lift a finger to be featured here." At least that's how I'd pitch it.
Personally, this might be a rather extreme idea, but I think there should be a rule/feature that prevents people from uploading art unless they've favorited plenty of other art on the site, maybe about 100 pieces. Getting people to lurk a bit and get to know the expectation of art quality, be they self-uploaders or not, will intimidate the less enthused ones, and challenge the people willing to rise to the occasion to actually put some work into it. It could stem the tide of new and weak uploaders a bit.