IMHO it would probably have been patently impossible for Early Medieval HRE to develop a strong central regime. Doubly so if it managed to take over Northern Italy. After all, it took realms with rather less geographical barriers for communications and markedly less fragmentary political landscape, such as France and England, some half a millenia to impose a working degree of central royal authority on their component parts - and at that point for example economic structures and much adminstrative praxis was something very different from what it was in the 1100s, not in the least thanks to major shifts in military paradigms.

Put short, the HRE was flat out too large and heterogenous for effective central authority given the adminstrative competences available at the time, and the problems of communications imposed by distance and geography. It did not help a bit that the Emperor was a primus inter pares to a meaningfully greater degree than most period European monarchs due to the fact he was elected into the office by the major Elector Counts; nor were the stubborn Freiburgs or wealthy, powerful and well fortified Hansa cities of the Baltic coast exactly readily accommodating to interference in their business. And the knightly Orders kind of tended to do their own thing anyway, helped by that fact their financial and adminstrative abilities were something rather beyond the norm for the times - and that various lords and kings tended to be heavily indepted to them.

Northern Italy would just have added another recalcitrant, wealthy and powerful piece - and one behind mountains, and hence even more difficult to communicate with than the forest-surrounded German parts, at that - to the already nigh-unmanageable mess of mind-boggling numbers of little fiefdoms, townships, bishoprics, baronies etc. that all had diverse sets of agreed-upon privileges they knew inside out and guarded quite jealously.

On top of the blunt fact the Emperor had his hands tied by the military and political realities of the feudal system as thoroughly as anyone else, an issue further compounded by the remaining Pagan lands to the east.