Instead there’s this divide with crappy $1 joints (that are now a little more expensive) and good $4-5 slice joints that had to differentiate themselves. The workingman’s NY slice might not have long to live.
All that said, it sounds like maybe you’ve not had good NY pizza the way you’re describing it. The sauce and cheese should not be messy, it should be minimal. The fold is so you can walk down the street while eating it in one hand.