This is wrong. Wikipedia was working right from the start, even before adminship existed.
It only worked because the vandals hadn't found it yet. If you removed adminship from Wikipedia now, we would not be able to control the vandalism. Obviously, I have no way to prove that, but I'm confident it is the case.
You're also forgetting that Wikipedia is not the only wiki, and that MediaWiki is not the only wiki engine. Most other wiki engines do not have an "admin" user level at all, and last I checked the Pattern Repository was still working just fine.
The English Wikipedia is by far the biggest and best known wiki, and as such is the biggest target for vandalism.