This is an old argument that has several very long and complex fronts to it. Let’s start with your first question, “where does the break go?” and finish with an introduction to its ramifications.
The short answer is this, the break material goes into your fermenter because it falls out of solution due to the change in temperature. Hence, “cold break/hot break.” It’s chemistry stuff that will happen at certain temperatures.
Does it matter? At a micro biological level it certainly does, that cold break combined with the dead yeast cells will start rotting once it settles to the bottom of the fermenter. Okay, it sounds horrible at the micro level, but does it have macro implications?
In my experience, no. I run at least four-five beers on each yeast cake. By the time I’m done with a yeast cake, the first wort’s cold break has been rotting for 2-3 months. Does it affect my beer? I wouldn’t keep doing it if it did.
I approach it this way: You can either go Jamil or Papazian on this issue. If the concept of rotting cold break grosses you out, that’s cool. Do the Jamil thing and more power to you. My pretty acute taste buds prefer the taste of beers that have come off a rotten pile of last month’s cold break, so I’m strongly Papazian on this issue. Aside from taste, there are a lot of tertiary reasons for my allegiance to the happy go lucky camp that have to do with biology and the stuff that happens when things go rotten. Still, I frequently sojourn into the Jaimist’s hyper clean camp for specific reasons.
At the end of the day, it’s up to you to figure it out for yourself.