I suspect that the strap portrayed consists of the beaded applique being sewn to one single wider leather strap, rather than being a webbing between two narrow ones. Remember, the function of the strap is to carry a heavy bag; the applique is for decoration. The strap needs some flexibility, sufficient strength, AND AS LITTLE STRETCH AS POSSIBLE. In this case, a wide leather strap with Native decoration. (If the man wearing it is indeed Sir William Johnston, there is a strong possibility it was decorated by his Mohawk wife, the sister of Joseph Brant.)
Although heavy wool fingerwoven straps on fingerwoven bags existed here early on, by the early nineteenth century, at least here in the Southeast, pillow-ticking had become the more common backing, with the beaded cloth applique being sewn onto it, with either beaded or ribbon edging. This combination also met the three requirements I listed above.