That's not how the process works, at all. The reason for the majority of cost overruns is the government changing the requirements during every phase, which generally means COMPLETE re-engineering because minor changes usually mean ripple affect on many other components/processes.
It's really easy to sit back and Monday Morning QB these things but it's a VERY complex endeavor that most people simply cannot comprehend. Fixed price contracts wouldn't fix much of anything, though I guess in the short run it would mean less money being spent - but in the long term I'd bet it'd cost much more.