How do we polish an old damaged bathtub?
The easy to your question is Yes. Clearly the products have shown that this is not an issue that will be solved with chemicals.

Be prepared that the remaining porcelain might not have the thickness needed to polish it.

You speak of buffing and polishing, that is the correct answer. You need to remove enough material to remove the 'dingy' parts.



The long answer:

Porcelain is a glass

To over simplify, they could be making pottery, or china tableware, instead they make bathtubs.

They take the metal tub, cover the surface with the glaze, and bake it at high temperature that will melt the glaze into a glass, cool it, polish it and put it in a box that they ship.

Polishing removes material, anything that gets polished, gets thinner.

Polishing makes things smooth, to make things smooth,is to make them level.

To make it level you can either get rid of the high spots, or fill in the low spots.

Waxing fills in the low spots with a different material, which clings to the surface where you applied it. For a while, and then you need to do it again.

Polish removes the high spots.

Almost everything in a tub, causes friction, friction gets bits of something to get in something, even if at a very small level. Water, dirt, and skin are among the biggest problems in a tub. Hot water only makes it easier to remove the porcelain.

porcelain can also be chipped or cracked.

The only options are to not use the tub, or to reglaze it. If there is not enough heat to melt the original glaze, then it is, at best, a really strong painted coating.