The problem with linen/cloth Over-the-Shoulder Cartridge Pouches for the 18th century Military or Militia was things like worms, turnscrews and even flints (when not in leather wallets) would/did abrade, poke through, slice, or cut the linen/cloth quicker and wear them out much faster than leather. This is part of the reason the military preferred leather, along with some added protection of the leather against rain/snow for their paper cartridges and other things carried in the Cartridge Pouch. Even the standard issue Cartridge Box, that we know as a "Belly Box," only carried cartridges and maybe an oil bottle. Though the Cartridge Box body was made of wood, the flap was leather for the added protection against the elements.
Heck, haversacks that they did not carry such Iron or sharp/pointy items in were only expected to last at most a year and often had to be replaced twice a year. Also, they did not carry anything in them that would abrade/cut/eventually ruin them - except from rot.
Further, documentation for the FIW and AWI war use of linen/cloth Shot Pouches is few and far between and almost always at the beginning of a war or when units were quickly raised and outfitted. This normally suggests there was not enough leather to make proper cartridge boxes or pouches, or they were in too much of a hurry to have them made prior to when the new outfits had to leave.
As to the number of paper cartridges seen as necessary/required to supply a Soldier for the British Army and British American Armies (until the AWI when some became Rebels) there remains a lot of confusion about how many rounds they carried. FIW period British Cartridge Pouches may only have carried 18 to 21 rounds, BUT they were also expected to carry an additional 10 to 14 rounds in the Cartridge Box at their waist. By the time of the AWI, British Cartridge Pouches had been increased to around 28 cartridge capacity, so they normally/often no longer wore the additional Cartridge Box around their waist and often stored them against future need.
Now, the American Army in the AWI generally copied the British where they could, but since their supplies and resupplies were often no where near as good as the British Forces, then, yes they may have carried 13 paper cartridges or less. That did not mean they figured that was enough, it was merely having to "make do" too often with what they could get.
Gus
P.S. Oh, I'm sure that ANY American Soldier in the AWI who survived a battle and when/if they actually won a battle; was VERY Quickly ordered to grab a British Cartridge Pouch of the defeated enemy. They may or even probably removed any British Device from the Pouch, but they would have quickly so outfitted themselves and used the linen/cloth pouches for something else - like a haversack most likely.