It takes a special skill to properly make, form, install & use lead so that it doesn't loosen during use. All who can't master the skill must use leather instead.
So said the old Gent who taught me how to shoot flintlocks from his family legacy collection. He would change flints at the range, and used both lead and leather. Some of the old leather in use was dry, brittle and crumbled when removed or scraped off the cock jaws.
His rule of thumb: larger (wider) frizzens all used lead & that was what he used when replacing old flints. Smaller (narrower) frizzens all got leather. He had an ancient, wide roll-up pouch that contained both lead of varying dimensions, and also leather.
The leather was dense & stiff and looked a lot like old cardboard that was soaked in tallow or ball lube. Some pieces were stuck to the inside of the pouch. It had no cushioning property, but did form around smaller flints just fine.