First, it is a pretty solid document of the Enlightenment period. There are flaws, but the rights-of-man and the reason-versus-religion dimensions are good, and the overall structure of government is good (better than the parliamentary system by a long shot, in my opinion).

Second, it is a straightforward statement of values rather than a prescription of points of law. That simplicity is a big reason why it's been stable for so long.

Under the U.S. system you can have a conservative government or a liberal government, while retaining essential human rights in either case. There was plenty of pushback to FDR's proposals, and there is plenty of pushback to GWB's proposals. I think that the Constitution is one of the strong points of the U.S. system.

Now, maybe I'm a little myopic (because New York has a really pathetic constitution), but I can't think of any state constitution that is demonstrably better than the stodgy old Federal Constitution. Yes, I'd like to see some changes (in particular an amendment guaranteeing freedom from religion), but all things considered, i think it's mostly evolved into a sensible, workable system.  

There was a lot about the dead-on-arrival EU constitution that I liked, mainly the parts about social and economic justice, but as a system of government it was, imho, far less democratic and progressive than what the U.S. has. And that, I'm afraid, is what would happen here as well; we'd trade a quirky but fairly workable system for a technocratic elite-dictated bureaucracy.