I'm not a drug war advocate, but I'll answer anyways. At the time they obviously believed they did. I would agree for a full ban on alcohol that they would. However, the federal government has authority over interstate commerce, so they could probably have made it illegal to trade alcohol across state borders or something like that, but local breweries and whatnot could still operate within their own state so it wouldn't be a full ban.
I don't think this is right...at least with regards to the local part...I seem to remember a case about someone who grew vegitables in their garden in excess of some congressional statute...now you'd think this was local and not interstate commerce, but i think the court disagreed. Something about a local activity having an economic effect on interstate commerce (a term thats broadly construed).
But I wasnt paying attention much, nor do I think I have the whole holding of the case down...and it probably came up in the late 30s after prohibition came and went.