And who runs foreign policy? How do state secrets work? Is there still a president and a bureaucracy and cabinet ministers?
All of those seem tied to the Executive branch, whereas direct democracy would replace the Legislative (Congress in the US, Parliament in other countries). The current Executive follows the laws that Congress passes, direct democracy wouldn't change that.
I think a direct democracy will still need a president and congress to handle many aspects of governance, like military action, state secrets, etc. My hope would be though that a direct digital democracy would take some of the decisions away from congress and put those decisions in the hands of the people. But certain aspects of government, particularly the military and judicial system will always have to be centralized.