In the process of figuring out this problem on my end. I have noticed like others that NEW YouTube app does use LTE data even when the phone is on Wi-Fi. Now the question arises is this a flaw in the YouTube App or iOS6. I have no answer on that.
But I can say that I have looked at my cellular usage so much today, that I can say that I have noticed, that it also has an occasional "data doubling" flaw for the usage counter. IE: I checked and had 50mb usage. I closed the app, put the phone to sleep, picked it up 5 minutes later and checked again and it doubled to 100mb usage, with no apps running in the background.
So in retrospective there appears to be two problems at play at the same time. The phone and/or app lack of ability to transmit on wi-fi as it should by default; and a flawed cellular usage counter subroutine built into iOS.
My thoughts are I can't wait until the next iOS 6.0.1 is released to hopefully fix these issues, and to keep me from having to check my Verizon account usage every several hours, because of the fear of going over my family's shared data plan.