In the 18th century great effort was devoted to keeping the cost of an inherently expensive institution (a standing army) down, both for the benefit of the State and for the benefit (profit) of the colonels commanding regiments, who were allowed to keep any difference between the the government's appropriation for the regiment and the colonel's actual cost for furnishing it. (see, for example Bennet Cuthbertson, A System for the Compleat Interior Management and OEconomy of a Battalion of Infantry (Dublin, Boulter Grierson, 1768)). The principle reason why white leather was preferred for soldier's musket slings (and accoutrements straps, etc.) was, economic.
The proper name for the white leather used for these purposes is "buff" leather. It is made from the same hides used for other kinds of leather using a process called "tawing." In the tawing process the hides are cured with a solution of alum and salt rather than the solution of oak gall/oak bark/sumac leaves, etc. used for common "tanned" leather. The result is a stiff, dense, strong leather of creamy white to light tan color that is far stronger and more durable than conventional tanned leather. It lasts three to five times longer in service than tanned leather, and hence, over the long run, is less expense to the regimental colonel than tanned leather which had to be replaced more frequently. (Also, it looks "cool," which was important in an age where parade ground "spiffyness" was important.)
British infantry regiments used buff leather straps, etc. pretty much universally, except for the slings and accouterment straps of the light infantry companies, which were commonly of blackened tanned leather. Scottish infantry regiments (many of which were raised for service in the American Revolution only and were expected to be disbanded at its conclusion) generally used blackened tanned leather for their slings and accoutrement straps. In the 16th and 17th centuries "buff coats" made of thick tawed leather were used by some officers as a kind of light armor that was cheaper and more comfortable than steel while affording pretty good protection against edged weapons and pistol balls. (But not musket balls. I saw in an English museum many years ago a buff coat that was soaked with the blood of its wearer on one side. It obviously had not been up to the task.)
In the Continental army white/buff leather was preferred when it could be obtained. But tawed leather was mostly imported and hence scarce and more expensive than tanned leather, which could be produced in many local tanneries. So most Continental soldiers who had leather slings and accoutrement straps probably had brown or black ones. Surviving artifacts indicate that many Continental soldiers had fabric accoutrement straps, so it seems likely that there were fabric musket slings in use also.