"Surveys" of originals have turned up Federal trousers with none (lost or removed then or later), round cotton cord, cotton twill tape (a common tie found in drawers), and two that had twine.
IMHO, this can come down to "as issued," "as used," and then what may or may not have been done as field expedients to replace a needed one that had been lost. Leather NUG makes for a poor tie. If for no other reason than leather often can be hard to tie, does not stay tied for long, is hard to untie when wet, and dries stiff or hard.
IMHO, when "anything is possible..." if a Period lad needed one and nothing else could be found, a leather thong or strip of leather would work. What happened in the Hobby is that some modern vendors came to use leather thongs 100% of the time. And the possible exception becomes the rule.
A parallel would be using a riflle sling as as a waistbelt. Or a cartridge box shoulder belt as a waistbelt. Or a corn cob as a canteen stopper.
And that can quickly take us to Mental Pictures and how focused do we want to be on presenting or projecting a more common, everyday impression versus choosing the rare or one-of-kind. Yes, the rare when documented can work for that man, that unit, and that time but may not work as well when made universal and is seen on tons of lads. (A classic example is Captain Samuel Richardson and his jaguar fur chaps and holsters.) Here, a leather thong is far less "egregious" than jaguar.
Others' mileage will vary...
In gleichem Schritt und Tritt, Curt Schmidt
Not a real Civil War reenactor, I only portray one on boards and fora.
I do not portray a Civil War soldier, I merely interpret one.