Let's assume that Otto I, instead of restoring the (Holy) Roman Emperors' title, unused for 38 years prior to his coronation, would have opted for elevating the Kingdom of East Francia (later Kingdom of Germany) to the range of Empire of Germany and avoided any kind of involvement in Italian affairs (or at least, not south of Tuscany) disconnecting the investment of him and his lineage from the policies of any Pope (specially the Tusculans of his era).
There is little reason for Otto to avoid the title, and many reasons for him to acquire it. It gives him a sort of supremacy (or at least guardianship) over the Church, which was critical given the centrality of the bishops in Otto's regime and the Ottonian policy of exerting control over its neighbors (like Poland) through the extension of German (arch)bishoprics. It gave him formal justification for the suzerainty he had already assumed over his fellow kings. It also legitimated his control over the contested region of Lorraine, because "Middle Francia" had historically been part of the imperial domain given to Lothair. But even if Otto did decline to seek the crown for some reason, this would not prevent his successors from seeking it. Otto, after all, was not the first East Francian king to attempt an Italian expedition; Arnulf of Carinthia had done it (although he held Italy only briefly), and Otto's father Henry was supposedly considering an Italian excursion at the time he died.
In retrospect one might argue that assuming the imperial title and becoming "involved" in Italy contributed to the fragmentation of the German state, but that was not at all apparent at the time. Otto's power and prestige were vastly increased, not diminished, by his assumption of imperial authority, and it is difficult to see why he would turn it down.
The idea of leaving Italy half-conquered and fragmented is a dangerous one, as Otto himself learned. Berengar II at first seized power in Italy with Otto's tacit support, was subsequently forced to acknowledge Otto's suzerainty, and was shorn of the Veronese march, but as soon as Otto was distracted Berengar made a bid to recover what he had lost and expand his own authority. Any king of a "rump Italy" with any dignity or ambition is going to be looking for an opportunity to re-unite the kingdom and claim the imperial crown which was traditionally associated with the Italian realm. No previous post-Carolingian attempt to divide the kingdom or share the royal title had worked out very well. It seems to me that rule over Italy is an all-or-nothing proposition; one must either subjugate it and allow no competitors, or leave it alone.
