Oxalic acid is a very mild organic acid. It has long been used to remove rust stains from fabric and will not damage fabrics. It is poisonous so wash your hands after using it or use gloves. I'll bet if you read the ingredients in your Tidy Bowl, you will probably see EDTA (ethylenediaminetetraacetic acid or one of its salts). EDTA is non-toxic and is also used in foods like salad dressing. It's great for dissolving soap scum and dried minerals like calcium and magnesium but I'm not sure about removing rust (iron oxide) from parts. 10 bucks for less than a pound seems like a Tidy profit for the store.

Vegas Triker, chemist