I am assuming that if you are talking scraps, you are not talking about pieces bigger than a few inches, and no long strips - - the strips/strings would definitely cause problems for a rototiller, however, if the one person who composts (piles, waters, turns) is correct, even those will break down.
Considering the process of gardening, the "chemicals" in fabrics are probably the least of your concerns.
Newspaper ink has been vegetable based for decades now, so a good friend to the soil. As a matter of fact, if you have ever read a book "Lasagna gardening", you will see that she absolutely uses the newspapers WITHOUT tearing them into teeny pieces even ... and, they still break down.
The "solution to pollution is dilution" .... you put your fabric in the ground, you water the soil, the heat of the earth starts it breaking down, the water you use to keep your garden growing continues the "dilution" process ...
it really is all common sense ... birds have no reason to eat fabric unless you soak it in something the birds want to eat ... I watch them pecking around in my cow piles all of the time, and they seem to be quite capable of finding one little gnat in a plop of poo <wave> or, maybe that one piece of grain that didn't get digested in the manure or horse pucks ... they won't eat your fabric!