I don’t know why companies still use the cotton drawstring over the elastic. I guess some people prefer it. Part of customizing my shoes are replacing the cotton with the elastic drawstrings. I even use them in my soft shoes.
I see the ends of the elastic to the ends of the cotton drawstring, give it a few good rugs to test that the connection is secure, and pull the cotton drawstring out pulling the elastic through. The elastic gives extra security to her favorite shoes!