There is one problem with that approach: the test is very late in the process.
Pilots and crews are busy enough in the final few minutes before takeoff. You are talking about adding another technical check at that busy time.
If there is a problem detected what should they do? You need more procedures and to determine how to correct the weight distribution and the co-operation of the passengers.
The existing procedures should work. They failed in this case because there was a problem at checkin: the children were checked in as adults. Adding another test at that time would be much more cost-effective than adding a test minutes before pushback.