Soaps, shampoos and detergents fall under a bit of a weird section of federal law in the USA. If there is a cosmetic claim to the product (“moisturizing!”, “Deodorizing!”) the label must show all the ingredients and is under the FDA’s jurisdiction. If there is a drug claim (“cures acne!”, “antibacterial!”) then it, too, must label all it’s ingredients and is regulated by the FDA. If no such claims are made other than “cleans!”, then ingredients are not required (which is why things like true animal-fat based soaps are not under the FDA’s jurisdiction, but, falls under the Consumer Product Safety Commission instead.
Cosmetics kind of fall into this grey area too - the are not fully regulated, other than colour ingredients, and it is up to the manufacturers to prove that products are safe. This is generally made easy by the simple fact that the vast majority of pretty much any cosmetic or detergent product you can find are made up of combinations of the same ingredients anyways (“Not tested on animals? Great, but that’s only because you’re benefiting from the fact that L’Oreal did it in the '50s!”) The CPSC and the FDA, as well as the manufacturers themselves, work collaboratively to verify new ingredients and formulations and act on consumer complaints within the scope of their respective jurisdictions. The FDA and CPSC have the weight of federal law behind them, so it’s not really an empty threat.
Regulatory bodies do have access to the full ingredient lists (“spices”, “aromas”) but they are generally kept from the public to protect the interests of the manufacturer. Since pretty much any dye, aroma or spice is already a known entity with known biological behaviour, there isn’t much concern that your “reformulated” shampoo will cause problems that every other shampoo has failed to cause in the history of time.
As a cynical ex-chemist (or am I always a chemist, having been one once?), the fact that consumers don’t know what a “surfactant” is doesn’t really matter. They could stick more specific ingredients on the surface, but it’s not like most people have any more of a clue what “sodium lauyl sulfate” or “sodium laureth sulfate” is, or what’s the difference (which are the most likely surfactants in the OP’s shampoo). I suppose they could just say “coconut oil derivatives” but that scares people just as much. Derivatives, what? The pH adjuster is usually just citric acid.
The most common ingredients in shampoos are Ammonium chloride, Ammonium lauryl sulfate, Glycol, Sodium laureth sulfate, Sodium lauryl sulfate, Sodium Lauroamphoacetate, Polysorbate 20, Polysorbate 80, PEG-150 Distearate, Citric Acid, Quaternium-15, Polyquaternium-10, Di-PPG-2 myreth-10 adipate, Methylisothiazolinone.
Pretty scary, but those are just the chemical names and here is what they do.