Alcohol prohibition was one part of a larger progressive social wave sweeping the nation as a result of Christian spiritual revivals during the 19th century. Other causes that had beginnings in the same social movements were slavery abolition and women's suffrage. Generally these movements were protestant based and in the case of alcohol prohibition were often at odds with urban Catholics.

Alcohol Prohibition probably would have come sooner but during the turn of the century alcohol prohibition was seen by many german-americans as an attack on their german heritage so it was politically untenable but the Great War (WWI) changed that dynamic and it allowed the constitution to be amended.

As far as seeing that Prohibition would breed a black market its obviously a bit 20/20. We can look at our own times and see how the creation of a black market isn't enough of a motivation to end the prohibition of other intoxicants such as cocaine, marijuana and heroin.

America was more or less founded on the concept that the government wouldn't try to set the average person's moral compass, right?

Not really. The definition of what freedom and liberty are have changed significantly over time. America was founded on the principles of liberty, at the time that meant the ability to live ones life in a proper and moral manner. What we would call liberty, they would call license, the ability to live life in any matter you choose.

So if someone was living is a state of license then they were not experiencing true liberty as it was seen as by many people at the time. So eliminating the temptation of alcohol the government is granting the drunkard a greater freedom than he had ever previously known.

How did anyone think that prohibiting the sale of alcohol was a good or just idea?

support was widespread and from many groups for many reasons.

suffragettes supported it as alcoholism was seen as a great threat that attacked the wife and child though the husband. Brewers and distillers also opposed women's suffrage for this reason, further antagonizing women.

Industrialist liked the idea that their workers would be more productive if sober. Progressives felt that it would help uplift the lower classes. Granting them a truer freedom then the license they currently enjoyed.

Religions leaders claimed that with prohibition "hell would be forever for rent".

Nativists felt that it was a blight brought by immigrants and prohibition would keep America American, they also largely felt that they would not be subjected to it. Up standing citizens are not the problem, the Irish and the Germans are.

In rural areas it was supported as an anti urban policy, although surely they would make an exception in the law for homemade cider; the most american of beverages. It's those city folk who cause all the problems.

Brewers supported it if it only targeted distillers, and vice versa.

I mean, America was more or less founded on the concept that the government wouldn't try to set the average person's moral compass, right?

No, the Framers' assumption was that state legislatures would regulate all manner of behavior for moral reasons and the greater good of society, just as governments always had. They only carved out a couple of small exceptions (speech, private property), and even those weren't always thought to restrict state laws. Our modern notions of liberty are mostly the creation of 20th century jurisprudence, which extended Bill of Rights protections to the states, and the new "right of privacy" that was created in the early 1960s.