I can only comment on the 2 main tires I use on my mtb's, 2.1 Velociraptors and 2.1 Nevegals, all Kevlar folders.
The 'raptors are my choice here in the Pacific Northwet. OK all rounders but more suited to some slimy stuff.
The Nevegals get put on for dry conditions and work well for me. My choice for the North Rim, dry with some dirt in the woods and rock/loose rock when exposed out near the canyon rim.
I did the Rainbow Rim trail a coupla years back. Pretty much XC yerrain, not much gnar. I do recommend the ride even if just for the awe of the views. Start at one end and do the out and back for 36 miles total.
Mountain Bike Bill's experience was similar to mine:
http://www.mountainbikebill.com/AZ06NRGC1.htm