That could become a plank of the Progressive movement, that all able-bodied men should serve their country to develop a stronger sense of national identity. Combine that with moralistic concerns (keep young men busy so they stay out of trouble!) and a desire for greater militarization (promoted by Theodore Roosevelt or an alt-timeline version of him) and you could get this instituted.
Of course, you should expect some pushback. Many families won't want their sons drafted, many sons won't want to go, and the Jeffersonian elements of the political culture are not going to be too happy about giving so many bodies to the military in peacetime. The practice may not last given pressure from these groups.