So I gave both opening Scout and going straight through Tradition a try.
Pros:
-I had a nice amount of money flowing, and never had a deficit.
-I had five cities before I'd even started building Notre Dame.
-These cities are growing nicely.
Cons:
-Everything other than my capital is POOP. My other cities may be 4's and 5's, but it takes them 40 turns just to build a Shrine.
-My Scouts are whiffing a lot. Ruins are really not very reliable.
-By the time I got my Enhancer Prophet, there were THREE Follower beliefs left. I didn't even want to look at which Reformation beliefs were left.
-I missed more than half the Wonders I went for. The +Production Policy in Liberty is very strong, I feel, and more useful early game than the +15% Production to Wonders policy in Tradition.
Bottom line - I'm going to stick to opening Tradition, opening Piety, and zooming through Liberty. The free GP finisher does NOT reset the timer, and is very useful. However, I will probably go through Tradition after Liberty instead of Piety, which is what I've been doing, because I can live with not getting To the Glory of God if I can purchase GEs with Faith in the Industrial Era no matter what. However, I think I will swap my build order and build a Scout before I build my Monument.