It's impossible to fairly discuss fermentation temperatures in the (mostly) style agnostic manner in which it's being discussed here.
It's thoroughly proven that colder temps favor enzyme activity in dough and enzyme activity is responsible for generating flavor, so, generally speaking, cold fermented traditional yeast doughs are more flavorful than warmer temp fermented traditional yeast doughs- fermented for the same amount of time.
It's also been theorized that coldness might impact the texture of dough adversely, but, nothing's been proven in this regard, and, out of my own experiments with room temp and cold ferments, I haven't seen this. Cold constricts the gluten and makes a tighter dough, but, once you let the dough warm up, the rheology appears to be identical to a dough that's never seen the inside of a fridge.
Coldness, though, doesn't seem to play well with starters. It seems very easy to generate an excess of acid- which wreaks havoc on texture. Craig, when you curse cold fermentation in the context of naturally fermented dough, I'm on board. When you curse it for traditional yeast NP, sure... I don't have a horse in that race. When you start cursing it everywhere, though, then I've got to say "oh, no."
Josh, for NY style, don't throw away your refrigerator yet. Without the refrigerator, you're not going to see the same level of enzyme activity, and, for the same amount of time, you're not going to see the same depth of flavor.
For NY, and for NY only, I've done plenty of room temp (70ish) proofs and plenty of cold ones. Colder is always better- and far easier, imo, due to the toss it in the fridge and basically forget about it for a couple days approach, as opposed to setting up equipment to maintain a stable warmer temp.