The constitution doesn't create or mandate a patent system. It only authorizes Congress to do so, if it chooses to. At first the range of patentable inventions was a lot narrower than it is now. Drugs were not always marketed under patent protection. The push to make drugs patentable came mostly from the makers of quack medicines.
Here’s a great piece on the history of pharmaceutical patents
http://www.lloydlibrary.org/scholar/Patents%20-%20V8%20-%20I1.pdf
And it points to Thalidomide as the reason compulsory licensing via the FDA was passed.
Patents are not the problem.