I'd like to clear up one misconception before going forward: Rome did have numerous non-Italian emperors. Some of the most successful and well-known, such as Hadrian and Trajan, were from Hispania. The Severan dynasty was from North Africa, and Phillip the Arab came from Nabataea. So achieving high rank within the Roman army was neither impossible nor uncommon for those from non-Italian backgrounds.

That said, to unpack the question further, we have to take a look at how the Roman government and army operated under the Principate and its relationship with the Italians. First, all Roman senators were required to live in Italy by law from the time of Augustus onward. These are probably those pesky "Italian" aristocrats, even though most of them came from the provinces. The Senate's composition grew much more diverse over time, especially as senatorial families died out and were replaced. The issue became most pronounced under Claudius, who fought the Senate in order to grant senatorial positions to those from Gaul. His success resulted in the gradual widening of the net, with senators from every province eventually making their way to Rome.

Under the Principate, and particularly the Five Good Emperors, the paths of advancement for a Roman citizen differed depending on if he were an equite or a senator. Again, thanks to Augustan legislation, these were defined social positions: to qualify as an equite, one had to have property worth 100,000 denarii and 250,000 to become a senator. Not unachievable sums, to be sure, but not the sort of money that one could reasonably be expected to earn in a lifetime without exceptional luck or holding certain professional positions (such as being a tax collector in a wealthy area or provincial banker).

The equites primarily filled in senior positions in the bureaucracy and military, where they would be able to develop the connections and acquire the funds to raise themselves to senatorial status. A good example of this is the Flavian dynasty, which rose from being the family of a disgraced Pompeian centurion to ruling the Empire within eighty years through clever marriages and lucrative career opportunities.

The senators, meanwhile, often served as governors, chief generals, and other similarly high roles. It should be noted that losing a seat in the Senate and failing to maintain the required level of wealth would result in demotion. Most senatorial families were relatively young at any given point in time, only one or two generations removed from being equestrians or common citizens themselves. That said, underneath the Principate, the Senate was clearly active and dynamic. So what changed?

Looking at the end of the second century, we find Septimius Severus and his descendants. Septimius Severus, upon taking power, had twenty-three senators put to death and opened the Senatorial posts to the equites. This was a major contributor to the disenfranchisement of the senatorial class, as their career paths and opportunities became completely cut off. He also returned the Senate to being a rubber-stamp committee as opposed to a somewhat functional legislature.

The reduction of the Italian aristocratic class to a group of idle, immensely wealthy landowners certainly seems to me to be an important contributing factor to the decline of Rome, since it reduced the number of avenues of advancement available to those seeking power and came alongside a weakening of Roman citizenship (particularly the protection against corporal punishment). The fundamental role of the Senate was to ensure that the most socially powerful men in the Empire were invested in its success and development. Without the Senate, the cloaked military dictatorship that had ruled since Augustus became open. As a consequence, the only thing that mattered was the loyalty of the legions rather than the civilian government, which reduced continuity between emperors and contributed to rising social unrest as the wealthy used their money to support usurpers.