I would unscrew the legs and look to see if another fabric is under it. I do upholstery and I cannot believe you cannot remove the old one. It got on there some way. If you could remove it, press it out and see if it is a full circle. Even putting pleats on the side could help. Because of germs, etc. that is on it, I would certainly remove it. You could cut it up cutting on the one side panel that is almost gone. Just do it and do not worry about it. I would not use wool or vinyl because of allergies. Good luck with it.