Actually, usually neither fiberglass doors, or corvettes are made out of the same material, and neither are what everyone thinks of as fiberglass. The term fiberglass reinforced plastic, is more accurate.
The faces of the doors are usually made out of a plastic material called GRP.
Corvettes have been made out of a material called SMC since the early 70's. SMC is a plastic material, that comes in sheets. It is a soft flexible product, until it is put into a press, and heated. There is fiberglass fiber in it, but it is nothing like the material made with resin and cloth, or mat, like boats made out of.
Taking a door to a body shop is an option, but usually an expensive one. I have sprayed many doors, and trim pieces, with automotive finishes, in my shop. No one ever likes the price.
I now spray doors solid colors, with products like Rustoleum satin enamel. Take it off, lay it down on saw horses, in the sun. Thin the paint 10%, and spray one side, one day, the other side the next.
I sprayed a fiberglass door for myself last year. I painted it white, and it looks just as good as if I used automotive paint, for a fraction of the cost.
If the OP insists on staying with the stained wood look, I would also go with the ThermaTru kit, as Jinman suggested.
As stated, the old finish must be removed. There is nothing you can put over the top of a finish, to make what is under it, adhere. If any part of it has peeled, the rest of it will probably peel too.