I just called them today and they basically confirmed what you said. They are taking unverified apps, but it's got to be complete. I am having the exact same problem as you (submitted early July.) Everything has been in for weeks, but no MCATs. Since they have the preliminary app, they think it's complete and tell me that it's complete. So now I think the file is complete, but when they go to look at it, there aren't any MCAT scores on the AMCAS app, so it just sits there for another couple of weeks until I got worried enough to check on it. I was not notified that my MCATs were missing either.
Anyway, I am having to fax/mail copies of my MCAT scores to individual schools now. Unfortunately, there are some that just don't take unofficial ones, so I just have to wait some more for those to come in.
Since you had problems with this school, I would recommend checking on the other schools you've applied to and find out if they have your MCAT scores, even if you've been told your app is complete.