It seems as though today, for “serious” applications (e.g., seatbelts, industry, military), webbing is preferred, and leather straps dominate only in areas where aesthetics are more important (e.g., civilian belts). It also seems like before maybe the late 19th century, leather was pretty much exclusively the material used for straps, irrespective of how demanding a role the straps were expected to play or the budget for the project. (Are there earlier uses for fabric webbing that I missed?)
Modern straps are often made of materials unavailable in the past (e.g., synthetic polymers like nylon), but it seems, from my reading, that the leading edge of displacement of leather by webbing occurred before the availability of synthetic polymers, with cotton or flax webbing.
Was flat webbing -- even natural fiber webbing -- a new technology in the late 19th century or so? Was there some weaving pattern that was uniquely strong that didn’t exist until then? Or some new buckling system that webbing straps more useful?
I guess what I’m wondering is why webbing started being used in the late 19th century or so, rather than either later (e.g., in the 1930s or so with new materials like nylon creating alternatives to traditional cotton/flax) or earlier (e.g. Roman fabricae making armor with webbing straps rather than leather).
Girth webbing straps do go back much further as use for re-inforcement in things like upholstered furniture, ship's sails, military tents, etc. I admit my knowledge of these items only extends back as far as the 14th century, but it was common use by then and "narrow goods" were woven in ancient times. Made of hemp or flax fibers, it can be extremely strong. Straps on items which are carried most often follow what the item itself is made of. A canvas knapsack from the Revolutionary War could have stitched canvas straps, a leather valise from the same time will have leather straps, for example.
It's not so much that the webbing takes on great advances, though in terms of industrial use all fabric changes drastically in the early 20th century. Instead it's the cost and quality of leather which changes. Leather today is considered a luxury item, example of car interiors, while in centuries past it was a cheap option (with exceptions of certain special leathers of course). Much of our leather comes from bovines, but today they are most grown to be as large as possible as quickly as possible. Much like your body would get stretch marks from such growth, the cows skin does as well, thinning out and loosing elasticity. This makes for poor leather. Loss of techniques for tanning and replacement with chemical chromium tanning that also produces a faster but lower quality of leather adds in as well. Webbing did not replace leather, but has been used along side of it for different purposes (leather has stretch, girth webbing does not, leather has "finished edges", girth webbing needs to be stitched to keep from unravelling, etc). Leather usage has just declined over the last century.