1. If the concern here is that persons that are eligible to serve also ought to be able to drink, then the most sensible way to resolve the inconsistency is to lower the drinking age.
2. The practical effect of raising the enlistment age to 21 is to deny the military its primary source of recruits (the average age of a recruit is 20 despite the enlistment limit being 35, meaning recruitment is heavily skewed toward the precise age bracket this change would eliminate). This seems unwise given the military is an AVF (All Volunteer Force) as is and given that it has been hard-pressed to meet recruiting goals this last decade (evinced by the fact that certain branches began letting in more felons, raising the enlistment limit to 42, etc).
3. Howard, I fail to see the sense in your proposal. You would create a class of military personnel who linger on the payroll for 3 years but are exempted from combat. What exactly would their purpose be? For goodness' sake, it doesn't take 3 years to train a soldier, and all the while these individuals will simultaneously not be contributing to the private economy plus will be of limited utility to the military. What argument is there to justify this extreme inefficiency?
4. If cost is a significant concern, the following should be kept in mind: defense spending entails spending on overseas contingency operations, the much-maligned military industrial complex procurement system, the training and retention of troops, and war-specific spending on troops (e.g., combat pay and disability benefits). There is lots of room here to lower costs without limiting the recruitment pool: for example, winding up the ridiculous wars will shrink contingency spending plus war-specific spending on troops, while reforming the procurement system in order to minimize waste and increase competitiveness also ought to cut costs significantly.
5. tl;dr - The inconsistency between recruitment and drinking is easily solved without affecting recruitment; raising the enlistment age would damage the military's effectiveness; it doesn't make sense for either the private economy or the military to create a class of recruits exempt from combat; costs can be significantly reduced without affecting recruitment; and, if the average 18 year old can figure out how to live on his own, choose a college to go to, find a partner, etc. then it is probably ok to grant him the right to serve (especially since service presents a viable route to cash/benefits comparable to the private sector -- something especially relevant in a difficult job market).
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