I have done them in leather, fabric, and in pleather. Your specific rubrail looks from the pic like it is not padded? In that case you have to scrape the glued goo off and get it really clean first. Then a new cover glued with contact cement.
If I can't see and it is padded, then upholstery shop as Pucho mentioned has the padding and the material.
From experience, there is a stiffness/durability continuum you have to walk. Very durable material is very stiff and does not finish well at the ends ie. school bus seat covering. Soft material looks great but does not last long. Fabric and leather went into private studios that do not get commercial use.
If there was padding, then the newer paddings are much better and stiffer. Less off-gassing too.
Sometimes the client has chucked the whole thing in favor of some nice wood, although it is not as comfy as a, err, mustache.
Mike