Perhaps the nearest modern equivalent to the woven yucca sandal of the Anasazi is the Teva sport sandal.
In the 1970s a Grand Canyon river guide named Mark Thatcher was frustrated that his clients either ended up with waterlogged tennis shoes and soggy feet or spent the day taking shoes off and walking barefoot on river rocks if they wanted to keep their shoes dry. The high-performance flip-flops worn by the guides kept feet well aired and were the envy of their clients, but didn't always stay on well or hold up during back-canyon hikes.
After the use of Velcro hook and loop fasteners became popular, and after a number of experiments, Thatcher came up with a prototype rubber thong sandal with an easy-open ankle strap and shopped it around. Those sandals spurred his initial success and inspired him to take a closer look at how sandals had been made and used by the Anasazi, who had to deal with the same sort of terrain the river guides and tourists did.
Anasazi sandals incorporated a toe loop in the center of the foot, putting the thongs of the shoe between two different sets of toes and circling them together, all of which gave lateral stability to the shoe. That, coupled with the ankle strap, really seemed to be the clincher for sport sandal success.
Though the Fremont's hobnailed leather moccasins may have had their uses in their terrain, the Anasazi's yucca sandals have proved sport shoes for the ages.
For more details, read Treading in the Past: Sandals of the Anasazi, a catalog of the Anasazi sandal collection at the Utah Museum of Natural History, edited by Kathy Kankainen and published by the University of Utah Press in 1995.