Yes, you are correct in that it only covers ratings after your private. Also, you are required to attend a Part 141 school that is VA approved. Part 141 schools have a syllabus that is more structured than a Part 61 school.
The amount the school charges doesn't matter. You get a certain amount of money total for your GI bill. When you use it, it's gone. There is enough in there to get your instrument, commercial and multi though.
It is a great deal though. I used it for my instrument and multi. If you already have the 250 hours TT it is not worth doing 141 for the commercial, because a large portion of the cost is building time.
One tip. I applied with Sallie Mae to pay for the rest. Because of the lag time, you have to pay up front before you get your checks from the VA. I would have paid for the flight training with Sallie Mae, then the check would come in from the VA, and I didn't send it to Sallie Mae. So, the VA paid for 60%, but I still have loans for most of it.