So, wouldn't this make the draft unconstitutional?
Gut answer: No, because Congress has the power to raise an army. I'm not sure why at the moment I think that this trumps the 13th Amendment, but I do.
In 1918 (WWI), a number of cases were filed testing just exactly this question. SCOTUS heard the issue in Arver v the United States (merged with five other cases).
The Court ruled:
As we are unable to conceive upon what theory the exaction by government from the citizen of the performance of his supreme and noble duty of contributing to the defense of the rights and honor of the nation, as the result of a war declared by the great representative body of the people, can be said to be the imposition of involuntary servitude in violation of the prohibitions of the Thirteenth Amendment, we are constrained to the conclusion that the contention to that effect is refuted by its mere statement.
With the advent of such warfare, mobilized property in the form of equipment and supplies became as essential as mobilized manpower. Mobilization of effort extended not only to the uniformed armed services but to the entire population. Both Acts were a form of mobilization. The language of the Constitution authorizing such measures is broad rather than restrictive. It says 'The Congress shall have Power * * * To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years; * * *.' Art. I, 8, Cl. 12.3 This places emphasis upon the supporting as well [334 U.S. 742 , 756] as upon the raising of armies. The power of Congress as to both is inescapably express, not merely implied. The conscription of manpower is a more vital interference with the life, liberty and property of the individual than is the conscription of his property or his profits or any substitute for such conscription of them. For his hazardous, full-time service in the armed forces a soldier is paid whatever the Government deems to be a fair but modest compensation. Comparatively speaking, the manufacturer of war goods undergoes no such hazard to his personal safety as does a front-line soldier and yet the Renegotiation Act gives him far better assurance of a reasonable return for his wartime services than the Selective Service Act and all its related legislation give to the men in the armed forces. The constitutionality of the conscription of manpower for military service is beyond question. The constitutional power of Congress to support the armed forces with equipment and supplies is no less clear and sweeping. 4 It is valid, a fortiori. [334 U.S. 742 , 757] In view of this power 'To raise and support Armies, * * *' and the power granted in the same Article of the Constitution 'To make all Laws which shall [334 U.S. 742 , 758] be necessary and proper for carrying into Execution the foregoing Powers , * * *' the only question remaining is whether the Renegotiation Act was a law 'necessary and proper for carrying into Execution' the war powers of Congress and especially its power to support armies. (emphasis mine)
While the Draft is clearly consitutional, I wonder if requiring people to join the Peace Corps or perform volunteer work would be constitutional. That is occasionally proposed in Congress.