Straps, belts, slings etc. were property of the ordinance department and had to be accounted for by company officers.
Also, a tumpline or short roll would probably be extremely cumbersome in close quarters. It did happen but was certainly the exception.
Even in that photo from Five Forks, there's plenty of knapsacks. There was no shortage of imported or domestically produced knapsacks in the Confederacy
Respectfully, a short roll would take up no more room than a knapsack and they can be adjusted to ride almost anywhere. I'll post pictures of an original private purchase sling from my collection, if I can find it.
Regarding the Five Forks photograph, I have blown this up many times and have only been able to ascertain 3 maybe 4 knapsacks, so the claim of plenty, is a bit too optimistic IMHO.
Existing Confederate made or imported knapsacks are fairly rare, especially compared to their US counterparts.
CS leather became a supply issue in the last year of the war, proven by the extant oilcloth belts, slings, cartridge and cap boxes. The amount of leather required for a knapsack was fairly substantial regardless of pattern. I am unfamiliar with pack made completely with oilcloth, slings and all.
I believe comfort and expedience were the driving factors of this choice of blanket carry.
Edit: Found the sling, this is called a coat or blanket sling and was marketed mostly to officers by Schuyler, Hartley & Graham. Made from the same type of leather as found in officers' belts, it had not faired well over the years as you can see. The sling shows no signs of ever being used.