Is it bulk fermentation or the balling after some extended period that makes the most difference?
Reif, it's the act of reballing after a period of bulk fermentation that makes the difference. This is b/c the longer dough sits the softer it gets as the enzymes break down the gluten matrix. Reballing rebuilds the strength in the dough and gives a better crumb. But this depends on several factors though. Getting a better crumb depends on not reballing too much, and also depends on not using a too low hydration with a too strong flour.
John, as an interesting point and side experimentation, I am willing to wager $ that if you were to make 2 batches, ball batch A straight from the mixer, ball batch B 12-24 hours later, AND reball a few dough balls gently from batch A at the time you ball batch B, that you will get a nearly identical result.
Hope that wasn't too confusing. In other words, keep your routine now, but just reball a few of the balls (after 12-24h of CF) that were balled straight from the mixer. Again, per your other thread, it's not specifically in the time frame per se. It has more to do with the building and rebuilding of gluten. The balance of dough strength. B/c of your year long experimentation, you have built up the necessary feel for the dough. You know just the right amount of balling the dough to get your desire results.
Here is another experiment to do that will shed more light on this topic. Make a batch of dough and bulk ferment cold. After 12 hours divide and ball all the dough as you normally would. To a few or half of the dough balls, do twice the number of folds during the balling phase. Or ball it really tight. Ball it until you can't ball it anymore or until you see see the outter skin breaking. Now allow the dough to rest another 12 hours or whatever your usual routine. Bake both up and post the results. Again, per my other post, we can definitely overball the dough as well. Per my experiments, great texture depends on balancing many factors: proper dough strength relative to hydration and flour type and proper bake time.