After his army had been smashed at Stilo. Otto II wasn't there at the suffrage of the Lombards, quite the opposite. He came south after Pandulf Ironhead's death and the appearance of an Imperial army was enough to cow the Lombards and force their (temporary) recognition of his authority as Emperor. It was only after defeat in pitched battle that everything fell apart for him. Which wasn't an atypical example of German expeditions south of Rome
Nope. This was before Stilo.
Otto II had come down precisely with the purpose of joining up with Pandulf Ironhead and conquering Byzantine Apulia. But Pandulf died in March 981, when Otto was still in Rome. While this ought to have put an end to the Ottonian plan, Otto II decided to insist, recognized Pandulf's two sons - Landulf IV and Pandulf II - as his heirs and proceeded on his campaign invade Byzanine Apulia in September. But the Lombards didn't agree. Still in that same month, Manso of Amalfi invaded Salerno and Pandulf of Sant'Agata captured Benevento, and drove out the Ottonian-backed princes. Otto II, who received the news while marching in Lucera, had to call off his invasion and return to try to take them back. He failed. His mighty army laid siege to Salerno in October, to no avail. By January, 982, Otto II capitulated - he was forced to give up the sieges and accept Manso's hold on Salerno and Panulf of Sant'Agata's on Benevento.
This was six months before Stilo (July 982).
So, Otto II, at the height of his strength and power, with all the king's horses and all the king's men at his disposal right there (arguably the largest German army ever seen in Italy) couldn't bring the plucky Lombard principalities to heel, i.e. Otto II was effectively defeated. After Stilo, of course, forget about it. He couldn't even flee through their territory. He had to go by ship around, to Capua.
Now I'd fully agree that "Benevento (and its Capua and Salerno fragments) was never under anyone's control but its own" but that wasn't for lack of trying on the part of the HRE. My point is that the German Emperors, the Ottonians at least, did consider themselves to be the rulers of all Italy and repeatedly attempted to assert their persumed rights in southern Italy. That they failed in this (brief periods aside) had nothing to do with constitutional niceties and everything to do with distance, competing powers, etc
Well, the Ottonians did not think too much about the legal t's and i's. They were usurpers, remember? They commanded very little loyalty and recognition among the traditional nobility. Heck, the Ottonians were hardly recognized in Germany itself. They were not Carolingians, they were not Franks from the Frankish heartland. Otto I didn't even bother with the formality of being crowned King of Lombards in Pavia (as all other kings were). The great Lombard-Frankish nobility, up and down Italy, rejected them. In their eyes, there were native dynasties which had greater claims as the rightful Kings of Italy.
Otto had next to no interest in Italy. But he had a lot of interest in the pope. Because the true power base, the muscle, of the Ottonians was the church - bishops in particular - and he needed the pope to ratify his appointments and deployments and erect bishoprics to dominate the eastern German borderlands against the Slavs.
Ottonians knew they were illegitimate usurpers, so there was no point playing nice. Ottonian policy was not to impress the traditional nobility, but to destroy it. And the bishops were their chosen battering rams. They tore down all the great nobles, broke up all the great duchies and marches, and wrote gigantic grants of property over to the bishops, and even (an Ottoninan innovation) wrote over the judicial and military duties of counts over to the bishops. This was also true in Germany, but it was particularly true in Italy where the Ottonians had no legitimate claims and near-zero local support.
(Yes, the Ottonians managed to cultivate a few native loyalists, usually lesser knights of low origin, who made out like bandits in the feudal spoils the Ottonians were handing out like candy (such as the hitherto unknown houses of Canossa, Este, Montferrat). But far and away, the church, the bishops, were the beneficiaries of Ottonian policy. The empowering of bishops with lay jurisdiction was the groundwork of the "communal era" that would follow the Ottonians, the fragmentation of northern Italy into a gazillion little bishop-led urban republics.)
However, not in southern Italy. He imposed not a person there. Pandulf Ironhead's dominions, great as they were, were unmeddled with. This was partly because he was the only great magnate in Italy who collaborated with Otto, and profited from the Ottonian rampage across Italy (Pandulf's rewards were Spoleto and Camerino). But it was also because it was partly thought as papal land. Indeed, when Otto was still in Pavia, it was the pope, by himself, who tried to opportunistically invade Benevento (Pandulf saw him off).
Otto was very intent on pleasing the pope. His bishops-and-bishoprics policy depended on the pope's cooperation. And the crown of emperor was what he really wanted to boster his usurpation back in Germany. The 962 Ottonian treaty with the pope confirmed everything the Carolingians had ever even hinted at giving them, and the 967 treaty went even further, giving the papacy the great exarchate of Ravenna (which the Carolingians had taken away).
Otto wasn't doing this because he thought of himself as a proper "King of Italy". He was doing this as a usurping thief, knowing he was a usurping thief, divvying up spoils to buy loyalty he did not legitimately command or deserve. But he didn't much care to actually rule in Italy. He had no claim there. There were native dynasties. Yes, he was concerned about snuffing out the prospect of German rivals fleeing to Italy, raising power there and coming back to challenge him in Germany. But Otto's interventions in Italy were all at papal request. Popes asked him to come, for reasons to do with toppling one Roman faction or another, or saving the pope from the Dukes of Spoleto, the perennial wolf at the gate. He was doing the pope favors, which the pope gratefully returned with the all-important bishoprics and the crown of emperor he craved. If not for opportunities to exact these papal favors, Otto wouldn't have bothered to cross the Alps.
To that end, his primary concern for breaking up the north was to clear the path to Rome, so that no magnate could block it. It was Rome he wanted to get to, south of Rome was not so interesting. Leaving the great southern duchies intact was fine - indeed, they were helpful to keep the Byzantines away from Rome.