I would even go so far as to draw the weights from a hat except for the final matches that could determine top team placements - leave those select matches for the very end.
What you suggested is NOT within the rules and cannot be done as such.
What is in the rules is that a random draw may be done in dual meets and for championship finals at both regular season and state championship tournaments. When a random draw is done only ONE weight class is drawn and that is the starting weight class, every match after that follows in the normal order and loops back through. For example if 140 lbs is drawn than that is the first weight class to be wrestled, followed by 145, 152 all the way to 285 and then back to 103. You can't just pick certain weights you want to do at the very end.
I am of the opinion that the starting weight in the finals at state should be determined by a random draw myself for the exact reasons that Will mentioned which is that it would keep people in the stands and give all these young athletes a chance to shine in front of everyone.
