I think you overlooked Section 2 of that amendment:
The Congress and the several States shall have concurrent power to enforce this article by appropriate legislation.
Congress was given the power to make all the necessary laws, but only within the context of alcohol being banned. If the amendment simply read "Congress shall have the power to legislate a ban on alcohol", it wouldn't actually have any effect on its own - those who supported it would have to get another law passed through both houses of Congress, and then a future Congress could trivially repeal it. By making the ban explicitly part of the amendment, it "permanently" locked in the ban that the temperance movement was looking for.
Additionally, the 18th amendment is structured the same as the 13th and 15th amendments, so there was precedent for this form:
- Section 1: Ban [slavery | racial voting restrictions | alcohol]
- Section 2: Congress can legislate about it
The 16th amendment could be is a counter-example, where it simply gives Congress the power to write new law, but there the amendment was specifically overriding one of the original Constitutional provisions, rather than adding new ones.