I understand. The rules as I've edited will play for storage until your opponent has 24VP more than you, THEN you will pick up the cards, BUT only if it's your last NV in hand. If not it'll set aside an extra card for when you do draw (because you still can, because you have at least 1 more NV in hand). Then after drawing, it'll be safe to assume you've drawn a few NVs. The rules will then proceed to set aside a few extra cards before finally drawing them again with the last NV, as opposed to the previous rules which would leave them on the mat. When you've played all possible NVs to draw as many cards as you can, the rules will then play all of your bridges.
That's what I meant anyway.
Edit: For example, a quote from danshep's game above, with my changes in bold which I believe would be strictly better than what actually happened (he still won):
— testinganstrategy's turn 15 —
testinganstrategy plays a Native Village.
... getting +2 actions.
... picking up 18 cards from the Native Village mat.
testinganstrategy plays a Native Village.
... getting +2 actions.
... drawing a card and placing it on the Native Village mat.
testinganstrategy plays a Native Village.
... getting +2 actions.
... drawing a card and placing it on the Native Village mat.
testinganstrategy plays a Native Village.
... getting +2 actions.
... drawing a card and placing it on the Native Village mat.
testinganstrategy plays a Native Village.
... getting +2 actions.
... drawing a card and placing it on the Native Village mat. picking up 3 cards from the Native Village mat.
testinganstrategy plays a Bridge.
... getting +1 buy, +$1, and reducing all costs by $1.
...
Now, obviously, I don't know what was on that mat, and maybe he set aside an extra card because the 3 set-aside cards were useless anyway. But had one of them been a bridge, for example, drawing them would be strictly better than setting aside another card which would go unplayed.