Even in the scenario you suggest, I think there is a good chance you can.
If you take the podcast at face value, and take your assertion on how US Mint sorts the coins, even than after a while companies like PCGS should be able to say a box contains good coins.
US mint surely have 500 of presses or more to press 10 to 50 million ASE, and how many strikes people a die is considered worn? Say after 500 presses of 5000 it starts to lose its edge.
So in day one, there would be 250,000 coins where the average is very good. Even if they are stored and boxed randomly unless US mint intentionally shuffle the coins they will be boxed and go to the warehouse as a batch. Plus chances all the dies are swapped at regular intervals too, meaning another 500 boxes of coins that contains a good chance of fresh strikes.
Now when a monster box comes in serial number 0000300 than boxes around that number are also likely to be good, so if NGC or Pcgs were monitoring serial numbers, they could guess if the box with a xxxx number is good or bad.