After mulling your answer over some more, it strikes me that the Federal Govt of the Progressive era didn’t really need a strict Constitutionalist justification for its involvement in anything; they already had precedents and SCOTUS interpretations that were giving them a lot of flexibility even then (albeit with some resistance from the SCOTUS).
So then the question is why in this case was a special amendment utilized? I had originally thought it was from some sort of constitutionalist sensibility that was a ‘check’ on federal imperialism but that doesn’t sound right anymore.
My new understanding is that it is because this was a bottom-up sort of initiative as do-gooder church groups and teetotalers petitioned enough States to get Prohibition of booze accomplished at a state level, then those same groups within the Federal system thought the country could now foist a federation-wide prohibition so that states that allowed liquor (wet states) would not be open-sieves to funnel illegal booze into dry-states. It was somewhat easier to put border control on the Canadian border than on a sister US state. According to Wikipedia, the war with Germany and the political-marginalization of German-Americans really helped remove the forces opposed to Prohibition from the equation.