I think a more powerful argument, rather than the military service one, would be "How can you trust someone enough to let them have a say in how the country is run, and yet not trust them to be responsible enough to have a drink?"
It's shit like this that gets Rick Astley elected president.
For the record, I'm not an American citizen.
Also, look at countries in Europe where the drinking age is lower than 18 (I swear to god, in Italy it must be like 12 or something judging by the number of infants I saw drinking wine in trattorias), they have a better culture of drinking than many Commonwealth countries (UK, Australia, NZ, Canada) and the US where it has become common place to drink to get drunk. In Europe this isn't the case, people drink for the flavour of the wine or apertivo or liquor that they are drinking, which is in my opinion what alcohol is for, not the side effects that come with excess.