They were absolutely an integral part. The auxiliaries often made up the bulk of the Roman cavalry and archers. Under the empire, they were as crucial as the legionaries and as well trained. It would have been impossible to control the Roman empire using only Italian based, or even just Italian led, forces.

Whether the influx of non-Romans was "bad" is really too subjective to answer. And really it is quite complicated. I'll say that early on auxiliaries were often made up of provincials. Provincials weren't citizens (until 212) but they were still part of the Roman empire. And many had been part of the Roman empire for a very long time. Rome would also uproot provincial auxiliaries and have them serve in another province to lessen the chance that a Roman auxiliary force would side with a local uprising.

However, from the very start people from outside the Rome (I use Rome to mean the entire empire) were crucial to the auxiliaries. Many "barbarians", especially Germans, would join the Roman auxiliaries. And could often rise high in the ranks. A famous example of this was Arminius who then broke with Rome and defeated Rome at Teutoburg and took their eagles. In that case, Rome was training their enemies.

Another case was the Illyrian tribes, who made up a great deal of the auxiliary forces and led a major revolt. But, on the flip side, those same tribes began to make up a large percentage of the Roman forces. And it was the string of Illyrian general-emperors, starting with Aurelian, who got Rome back on track in the third century. In that case, Rome's opening up their military to talented non-Romans saved it from oblivion.

Through out the first and second centuries (into the third), auxiliaries were very much Romanized although retaining their local flavor and customs. Non-Romans who rose through the ranks to lead forces would often be granted citizenship or high status. They wanted these forces to become as Roman as possible. That's what makes an auxiliary different from a mercenary. These were Roman troops who were being trained like Romans. However, that changed with time. Eventually, entire Germanic tribes were fighting for Rome under their own tribal leader or king to fight off other Germanic tribes. For example, the Huns fought for Rome against the Goths. Then the Goths fought for Rome against the Huns. Their loyalty was to their tribal leader not to Rome. At this point, they are more like mercenaries than auxiliary troops.

A key example is Alaric, leader of the Visigoths who sacked Rome in 410, who had been in the Romany army. But, even then, what Alaric wanted was not to conquer Rome. He wanted to be made head of the Roman army in the West. Most of these tribes (who were being harassed by the Huns and wanted to be within Rome's borders) would have preferred a place within the Roman military system than to be rulers of their own kingdoms carved out of Rome. So, some historians see it more as a failure of Rome to accommodate huge bodies of outsiders within their system, which they had historically been quite good at. Whereas other historians, see the mass migration of Germanic tribes within Rome's borders and in Rome's army as the problem. There is no easy answer. But, regardless, I think that entire process is different from the auxiliary forces, who had occasional revolts but were essentially loyal and Roman, even if not citizens.

So, yes, auxiliaries were crucial to Roman military success. This wasn't necessarily a bad thing. Yes, sometimes tribes within the auxiliaries revolted and turned against Rome. Sometimes those revolts were major. There are other examples aside from mentioned above. But legions revolted against Rome sometimes too. There were numerous civil wars led primarily by the legions. So, it isn't as though this was just an auxiliary issue. And a great many talented soldiers from the provinces or outside the empire were huge assets to the Roman forces. Towards the end of the Roman empire in the West, introduction of non-Romans into the military had bad results but the system was significantly different from what the auxiliaries had been before.