I agree with all of this, but I'd say that it was the 30 years war which destroyed much of the idea of a unified HRE.What we call "the Holy Roman Empire" was the combination of three of the successor states of Charlemagne's empire, Germany, Italy, and Burgundy (Burgundy-Provence or southern Burgundy). The situation with Burgundy was really complicated and it lost all its territory by the middle of the fourteenth century, so it was Germany and Italy that was important. And medieval Germany worked as well or better as any other medieval kingdom. It had common institutions, defending itself from external enemies and expanded its territory, and internal conflicts were the sort of dynastic conflicts that every kingdom experienced. It had post-Reformation religious wars, including a really bad one after 1618, but so did everyone else. Germany only ran into problems when Friedrich II of Prussia started undermining it.
Any state is defined and bound by it's common institutions, and HRE's institutions had suffered devastating damage by the time of the peace of Westphalia.
For instance, the Imperial Diet and the Electoral College began to be treated like we would treat General Assembly and the Security Council, instead of internal institutions where legally-binding policy would be made. The emperor, instead of being the head of the state, became more like Macedonia as the hegemon of the Hellenic League, or USA in the NATO.
I'd say of the 4 requirements you have put down, it can be argued that 3 were already true.The challenge, with a POD between the 800s to 1800s, have the Holy Roman Empire actually work. Make it the superpower of the continent, a centralized state, under one leadership, and the the Title of Holy Roman Emperor be powerful and prestigious, making it a worthy successor to the Roman Empire.
The one which could be different is the second one, but PoD would have to be pre wars of religion. No way any kind of centralization, built on the premise of a shared cultural heritage, survives 'Cuius regio, eius religio'.
I guess Ottokar II of Bohemia getting elected emperor seems a fair PoD. Habsburgs had to be way more diplomatic and respect the privileges of everyone to undertake any kind of centralization, especially given the way they ascended the imperial throne. They did very well in providing stability, but in the long run this could not convert to centralization.