I'm not quite sure how big the prospect of a surviving Ostrogothic Kindom of Italy is with a POD after the death of Theoderic. Theoderic had elevated Italy to become the strongest of the successor kingdoms and gotten quite close to restoring the Western Empire politically by almost forming a union with the Visigoths and allying with the Vandals. However towards the end of his reign all of his plans had started falling apart. With Amalasuntha dead I don't really see the Ostrogothic court remaining too stable. Theodahad was elderly and didn't prove to be a very capable leader.
Furthermore the relative religious tolerance under Theoderic's early reign was de facto gone by this point as such the Gothic position was pretty vulnerable even if the Romans are pushed out. They could potentially be victim to attacks from the Franks or Avars.
However if they manage to sort out these issues (Say the tolerance of Theoderic's early reign is restored/the Goths convert to Orthodoxy and they sort out the political problems) then Italy would for sure be much more developed and culturally Roman than it was for most of the Middle Ages in our timeline. Rome would not be besieged and depopulated twice, Mediolanum would not be razed to the ground, the remaining Senatorial Elite would not be killed, taken hostage or flee and the penninsula would retain a much more urban, wealthy and perhaps cosmopolitan character for a longer period of time. The Papacy would also be severely weakened as it would lack it's political independence.
The amount of butterflies this would create makes it very hard to imagine a long-term scenario. It's even quite plausible this could butterfly the Arab Conquests, which creates so many changes I can't even begin to imagine what the long term consequences would be.