...just looked at mine (78)
My dowel runs through a sewn sleeve.
I would not recommend stapling as there will be uneven tension at each staple and it could cause tears. I would recommend removing the assembly so you can have the sleeve re-sewn. You may be able to use a home machine with heavy thread, or you may prefer to take it to an auto upholstery shop.
Inspect the holes where the stitching has come out...it may be able to be re-sewn, or it may be beyond repair and need replacement if the sew-holes are showing excesive wear, which would cause new stitching to fail anyway.
Silver
*If the design has changed to some type of mechanical melt and press-together (fused) combination, I don't know what to suggest besides replacement. I wouldn't know of a glue that would hold up in a vehicle's interior heat.