Where did you see that they simplified rules for the Quick Start? It was my understanding that the Quick Start was the jumping off of featuring the newer features of the rules but wasn't the complete set of rules. As far as I know WotC just omitted some rules that weren't important to running that adventure.
Have you looked at the Rules PDF on the website? Those are, more or less, the complete rules for the system and you could pretty much use those to run a whole campaign.