It's not a big deal. Just drive the TOW.
In my opinion, the only question is in which direction do you go. We loved our Chicken -> Dawson trip in 2014. In the only skinny part (as I remember it), we were on the uphill side. Wife likes that.
That direction also let us stay in the government campground across the Yukon from Dawson and we could walk downstream and see those big ol' steamer hulks on the bank. We crossed on the ferry first thing in the morning.
I might have missed it but have any repliers gone off the road or even got stuck on a soft shoulder or are these "urban" legends.