The FDA regulates "drugs" under the very thorough Federal Food, Drug, and Cosmetic Act. Drugs, food and cosmetics that come under the jurisdiction of the act are quite well defined, as well as what and how the FDA is to regulate them. The statute doesn't say, "The FDA should regulate drugs." The statute defines what is a drug, what conditions a drug must meet in order to be regulated and how it is to be regulated. What causes any particular drug to be regulated is that it meets the definition and conditions that Congress established.
A good example is the definition of catfish, found within the act, "the term 'catfish' may only be considered to be a common or usual name (or part therof) for fish classified within the family Ictaluridae;..." No other fish can be regulated as a catfish.
The term "drug" means (A) articles recognized in the official United States Pharmacopoeia, official Homeopathic Pharmacopoeia of the United States, or official National Formulary, or any supplement to any of them; and (B) articles intended for use in the diagnosis, cure, mitigation, treatment, or prevention of disease in man or other animals; and (C) articles (other than food) intended to affect the structure or any function of the body of man or other animals; and (D) articles intended for use as a component of any article specified in clause (A), (B), or (C).
Furthermore the act goes on to state that the drugs have to be involved in interstate commerce.
So, while the statute doesn't list every drug that is regulated by the FDA, the statute gives a very clear definition of what a drug is and gives authority to the FDA to regulate it. But, what can be regulated is also very well defined. The statute lays out very specific prohibited acts that the FDA is supposed to regulate; these acts include: adulteration or misbranding, receipt of adulterated or misbranded drugs, false guarantees of what the drug does, forging, counterfeiting and a host of other items.
Even though the act doesn't list all approved drugs it does identify what list is to be recognized by the FDA, e.g. United States Pharmacopoeia.
The act was originally passed in the late 1940s and has been amended many times. Without the act the FDA wouldn't be permitted to regulate food, drugs or cosmetics. If the item doesn't fit the definition of a food, drug or cosmetic as outlined in the act then the FDA isn't permitted to regulate it. If the action isn't prohibited by the act then the FDA isn't permitted to regulate the action.
I would never argue that "drugs" as defined under statute is too broad and not specific enough because the act gives the lists from which "drugs" is to be taken. Why do you think the FDA is unable to regulate the late-night snake-oil infomercials? Could it be because those items don't meet the definition of drugs as laid out by the act? Or is it because the actions aren't prohibited? I think a layman watching the commercials would think that drugs were being advertised and they would be wrong.