That may have been a vinyl covered backing that is stitched along the sides or one molded from plastic. I have used old belts from thrift stores.
I restored an 14" RCA Victor metal cabinet TV for a friend who then repainted the aluminum shell, paying $25 for a custom stitched handle to be reproduced.
With all the must-do's we put into these players, like idler wheels ($$$), $25 is a bit over the limit for a perfect handle.