In the early 1900s, many American Protestants strongly supported Prohibition, but in order to outlaw alcohol (or anything), the only option Congress had was to wait for the States to amend the Constitution.
Congress obtained the power to lay and collect taxes on anything (including labor) in 1913 with the newly established 16th Amendment, and immediately used this power to place harsh restrictions on heroin and cocaine with the Harrison Tax Act of 1914. However, the Harrison Act did not prohibit heroin and cocaine, and it was overturned in 1925 in Linder v. United States when the Supreme Court ruled that the federal government didn’t have the authority to regulate medical practice.
This is why the old cannabis law was called the Marihuana Tax Stamp Act of 1937. Prior to this Act, cannabis was gaining popularity as a pharmaceutical- having been used for everything from pain to epilepsy to mental health disorders. Smoking cannabis was also becoming a popular alternative to alcohol, until Harry J. Anslinger began his propaganda campaign against it after Prohibition was repealed.
Unlike the Harrison Tax Act, the Marihuana Tax Stamp Act intended to completely outlaw cannabis by requiring a stamp to posses it. However, the only way to get the stamp was to bring in the unstamped cannabis; a catch-22 that made legal possession impossible.
Timothy Leary challenged this in 1969 in Leary v. United States. Leary claimed that because you had to possess the cannabis to get the stamp, the law forced self-incrimination. The Supreme Court ruled in his favor and the Marihuana Tax Act was repealed.
However, the next year Congress enacted the Controlled Substance Act of 1970 (CSA). This law [ab]uses the Commerce Clause of the Constitution, which says:
[The Congress shall have Power] To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes- Article I, Section 8, Clause 3
What this means is, Congress has the authority to make laws concerning interstate commerce, or commerce that passes across state lines. So if someone in California wanted to trade with someone in Nevada, Congress has the Constitutional authority to regulate this trade.
However, when it comes to intrastate commerce, or trade within a state, that is outside of Congress' legislative power. It was up to the states to decide various laws concerning intrastate commerce. They get this idea from the 10th Amendment:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Here is the beginning paragraphs of the CSA, and then I'll explain what this means:
PART A - INTRODUCTORY PROVISIONS
§ 801 Note Short Title...
...(3) A major portion of the traffic in controlled substances flows through interstate and foreign commerce. Incidents of the traffic which are not an integral part of the interstate or foreign flow, such as manufacture, local distribution, and possession, nonetheless have a substantial and direct effect upon interstate commerce because -
(A) after manufacture, many controlled substances are transported in interstate commerce,
(B) controlled substances distributed locally usually have been transported in interstate commerce immediately before their distribution, and
(C) controlled substances possessed commonly flow through interstate commerce immediately prior to such possession.
(4) Local distribution and possession of controlled substances contribute to swelling the interstate traffic in such substances.
(5) Controlled substances manufactured and distributed intrastate cannot be differentiated from controlled substances manufactured and distributed interstate. Thus, it is not feasible to distinguish, in terms of controls, between controlled substances manufactured and distributed interstate and controlled substances manufactured and distributed intrastate.
(6) Federal control of the intrastate incidents of the traffic in controlled substances is essential to the effective control of the interstate incidents of such traffic.
(7) The United States is a party to the Single Convention on Narcotic Drugs, 1961, and other international conventions designed to establish effective control over international and domestic traffic in controlled substances.
Basically, this means that manufacturing (or growing) and possessing certain substances (or plants) is usually done so through interstate commerce, so the law bans this. However, because it is "not feasible to distinguish" between substances manufactured or possessed in intrastate commerce, all intrastate commerce of these substances are also banned.
For example, if I grow an acre of cannabis to sell within my state, the federal government cannot be certain that the cannabis won't enter the interstate market, so that acre is a federal crime; despite what the law says in my particular state.
But it gets even better. In Gonzales v. Raich (2005), Angel Raich's six cannabis plants were destroyed by federal agents, despite the fact that California had passed legislation to allow Raich to grow her own medicine. Raich claimed that the cannabis was obviously used for personal use, so it wouldn't have any effect on interstate commerce. The Supreme Court ruled that it didn’t matter how many plants she had, nor what their intended purpose was.
The Court cited an earlier case from 1942, when a farmer named Roscoe Filburn had grown more wheat than he was permitted to. Filburn claimed that the wheat was for his own personal use, so it did not affect interstate commerce. In Wickard v. Filburn the Court disagreed, stating:
...that a factor of such volume and variability as home-consumed wheat would have a substantial influence on price and market conditions. ... Home-grown wheat in this sense competes with wheat in commerce. The stimulation of commerce is a use of the regulatory function quite as definitely as prohibitions or restrictions thereon. This record leaves us in no doubt that Congress may properly have considered that wheat consumed on the farm where grown, if wholly outside the scheme of regulation, would have a substantial effect in defeating and obstructing its purpose to stimulate trade therein at increased prices.
Now it doesn't matter if you have acres of cannabis or a joint in your pocket; it is "more than likely" you got that joint from interstate commerce, so it's illegal. It doesn't matter if you grow your own cannabis for personal use, because if you grow your own you're not buying it from the market, so your cannabis plant competes with the cannabis market. Replace Cannabis with tomato or chair, and you can see that Congress can ban absolutely anything they want for whatever reason they want, no amendment required.
So the reason why an amendment made more sense at the time is because Congress did not have the Constitutional authority to ban alcohol. When they tried other ways to ban substances, they were also unconstitutional. Now they can ban anything.