Barbarians were more influential in Rome's military system because there were more of them serving in the military than in previous centuries. There were no Celtic, to keep with your example, generals in part because there simply weren't enough barbarians serving in the Roman military at the time. By the fifth century Rome became more reliant upon barbarian soldiers, though the majority of soldiers would have still been Roman citizens (probably as many as 75%), this is what allowed men like Stilicho, who was half-Vandal, to rise as high as magister militum. While Stilicho and Flavius Aetius, whose father was probably a "scythian", were of barbarian heritage they still viewed themselves as being Romans, which probably helped them rise to the highest ranks the military had to offer.

As to why there were more barbarians in the army than in previous centuries, its partly because the Empire started to actively recruit barbarians. The Romans probably weren't in desperate need of more manpower, at least at first, but the fact that they started to allow larger numbers of Barbarians to serve implies that they must have had at least a small shortage. Emperor Valens, for example, allowed the Goths to settle inside the Empire as dediticii, which meant that they had to supply a certain number of troops to the army every year. One of the reasons that Valens did this was so he could have more men for his war against the Persians, and the Goths had at least 10,000 warriors when they crossed the Danube in 376. Another example of this is when Emperor Honorius settled the Visigoths in modern Aquitaine. The western Empire was also likely not strong enough to completely remove the large number of barbarians that had began to move into their territory, though this is not to imply that they were weak. Honorius' settling of the Visigoths in Roman territory is evidence of this, as they were, to a certain extent, their own independent state within the Empire's borders. Another example of this is Alaric, who of course rebelled against imperial rule. even though Stilicho defeated Alaric in battle in 402 and 403 the Goths were still able to remain inside the Empire. In fact Alaric even reentered imperial service for a time before beginning his second invasion of Italy. This likely means that the Western Empire either did not have the resources to completely oust them, or they still wanted/needed the manpower that the Goths could provide them. The Romans also began to lose control of Spain around 409 when the Buri, Vandals, Suevi, and Alans (who were actually Sarmatians rather than Germanic like the other three tribes), migrated into Spain, possibly at the behest of a usurper. The Romans of course did not lose control of the entirety of Spain immediately, but nonetheless it did hurt the Empire. Honorius' settling of the Visigoths in Gaul was probably an attempt to establish them as a first line of defense in case the Empire was faced with a significant threat out of Spain. In 451 Emperor Valentinian III even made a treaty with Theodoric I, which may have more formally recognized their autonomy, in order to get their help against the Huns who had invaded Gaul.

In addition to recruiting from those who had been settled inside the Empire, the Romans would also allow some defeated barbarians to serve as well, and others probably did not have very much choice. For instance when Radagaisus invaded Italy in 405/406 he had around 20,000 fighting men with him. After his defeat approximately 12,000 of these were drafted into Roman service. In 408 after Stilicho's execution, he was accused of trying to usurp the throne to place his own son upon it, the families of the Gothic troops were massacred. Tens of thousands of Gothic soldiers then deserted and joined Alaric. This incident shows that the Romans, despite having such a large number of Gothic troops in their service, did not feel threatened by their presence, and were not reluctant to slaughter their innocent families.

In addition to this I think it should be said that the Romans did not recruit more barbarians from a position of weakness. They were still quite capable and won several major victories in this period. Aetius defeated the Visigoths on three separate occasions before they became allies in 451. Theodosius I carried out a, somewhat, victorious campaign against the Goths from 379-382. Emperor Gratian defeated the invading Alamanni in 378, and then led a successful campaign across the Rhine as well. Roman troops still had better training and equipment than most barbarians, and the barbarians who did serve in the Roman military would have access to these as well. So the Romans did not recruit barbarians because they made better soldiers, though they may have been perceived as being more "ferocious", they were simply a large source of manpower. So its not as simple as saying that the Romans were "weak" but that they simply were not in a state to handle such a mass migration in so short a period of time, and the new potential source of manpower would have been very tempting. Men like Stilicho and Aetius were able to become magister militum because they were great generals, rather than a lack of good Roman officers.

Sources:

Halsall, Guy. Barbarian Migrations and the Roman West, 376-568. Cambridge: Cambridge University Press, 2007.

Heather, Peter Empires and Barbarians: The Fall of Rome and the Birth of Europe. Oxford: Oxford University Press, 2009.

Heather, Peter (2006). The Fall of the Roman Empire: A New History of Rome and the Barbarians (2nd ed.). New York: Oxford University Press

Marcellinus, Ammianus. The Later Roman Empire: AD 354-378. Translated by Walter Hamilton. New York: Penguin Books, 1986.

Orosius. Historiae Adversus Paganos

Recommended sources that weren't used for this answer:

Heather, Peter. Goths and Romans, 332-489. Oxford: Clarenden Press, 1991.

Gregory of Tours. The History of the Franks. Translated by Lewis Thorpe. New York: Penguin Books, 1974

Historiucs, Zosimus. Nova Historia. London: Green and Chaplin, 1814.