You also have to keep in mind that debates about Constitutionality arise over politically unpopular topics. It's a way for people with minority viewpoints to say: "well, whether you think this is good or bad, it is inconsistent with this larger scheme we all agreed to."
If you held a vote and asked people whether they thought people accused of terrorism, especially abroad, should have rights, you'd lose. I remember in college being the lone dissenting voice in a discussion of the merits of turning the entire Middle East into a glass parking lot. Ordinary Americans believe very deeply in the sovereign right of the American government to kill foreigners who do anything contrary to American interests.
Hence the resort to legalism. You can't convince Americans that it's good for suspected terrorists to be given trials, etc, so the best you can hope to do is convince them that they, collectively, bargained away the ability to hold suspected terrorists without trial.