This is actually a really important consideration that sometimes gets forgotten when people talk about the collapse of the Roman Empire in the fifth century. It is important to acknowledge that it was only the western half of the Empire that disintegrated and disappeared while the eastern half continued relatively unchanged. This forces any theory on why the western half collapsed to address why the eastern half survived, or the survival of the east can be used to poke holes in the theory. This immediately weakens popular theories like the Roman Empire collapsed because of lead poisoning/climate change/Christian morals/being too large to manage/any other broad and all-encompassing theory!

The reasons for the survival of the eastern Empire and disintegration of the west can be boiled down to a few key differences. Firstly, there were major differences in the economic strength of each half of the Empire. The eastern Empire included some of the longest developed economic centres of the human civilisation - Egypt, Mesopotamia, and the eastern Mediterranean coast. Greece had also had a developed economy long before many places in the west. The western Empire had rich natural wealth in parts of Spain and North Africa, and a very healthy economy in Italy, but much of Illyria, Gaul, and Britain did not have the same history of economic development. This meant the eastern economies were not only wealthier but also more robust and resistant to upheaval.

Related to the above point was the geographic advantages of the east. Constantine had created a 'new Rome' at Constantinople, which sat directly on a very narrow pass of the Bosporus. Especially with the development of the Theodosian Walls in the early fifth century this city became an impenetrable blockade for any barbarian groups wishing to cross from Thrace into Asia. The Huns were the most famous group, but also the Goths raided and ravaged Thrace and Greece but were unable to strike further into the eastern provinces because of this bulwark. Fortunately for the eastern Empire the late fourth and fifth centuries were a time of peace with the Sassanian Empire further to the east. This combination meant those economically developed territories of Egypt, Mesopotamia, and the eastern Mediterranean coast survived the fifth century unmolested and economically strong.

Conversely, in the west, there was no similar natural boundary to block the barbarians. Raiders were able to come down from the Scottish mountains and raid Roman Britain, and Germanic people were able to cross the Rhine river, which they did en masse in 406, and devastate Gaul. The Vandals, from whom we get the modern word 'vandalise', were part of this mass-crossing in 406 and they spent years devastating the economy of Gaul. They continued south and crossed into the Iberian Peninsula, occupying the province for around twenty years. I haven't spoken to him in a while but Katarn04 was doing a dissertation on the impact of this on the western Roman Empire, maybe he can chime in with some detail.

But after this the Vandals crossed into North Africa which was a turning point for the western Empire. North Africa, after Italy, was the richest province in the west and provided monumental amounts of food to the rest of the Empire, especially to Rome itself. The Romans spent a lot of resources trying to win North Africa back but failed each time and the economic drain was huge. And the Vandals weren't the only tribe to raid the Empire - the Franks, Suebi, and Goths all invaded the western Empire. This eroded the tax base from which the Roman Empire was able to recruit new legions and hire barbarian mercenaries, even from which it could pay-off barbarian raiders. This was not an instantaneous loss, because the Roman army was still effective into the mid-fifth century, such as when Aetius defeated Attila, but it was a blow that was felt strongly in the long run.

To deal more closely with your actual question, there were differences as well in how the two halves of the Empire dealt with the barbarian mercenaries. You have accurately identified that it was the lack of loyalty by these groups that were their weakening factor. How this lack of loyalty manifested in the east and the west was different. In the west there was a rise of the generalissimo. Beginning with Stilicho's appointment as regent of Honorius in 395 the west is ruled for the next 60 years by emperors who functioned as the puppets of strong military men. Meaghan McEvoy has framed this phenomenon as the emperors being allowed a 'ceremonial' role while the 'active' role of the emperor was commandeered by the generals of the army. This lead to a great deal of internal fighting and plotting - generals fought against each other for the supreme position, only to fall afoul of the intrigues of court eunuchs. Generals used personal loyalties and bribes to hire private armies of barbarian soldiers only for those same soldiers to turn on the Empire if circumstances made it advantageous. I think most importantly this separation of 'ceremonial' and 'active' roles of the emperors meant they could not take decisive action to rectify the situation. The Empire's lack of concrete succession policy meant its history is full of civil war, but in the end a strong leader always emerged and set about rectifying the problem. These ceremonial emperors of the fifth century were powerless to take such actions - in fact Valentinian III, in front of his entire court, dramatically drew a sword and suddenly murdered his controller, the general Aetius, only to be assassinated himself a few days later. No one in the western Roman world wanted a strong emperor, they wanted an emperor they could control.

This same situation did not come about in the east as strongly. In an ironic turn of events, seeing as Christianity was one of the first factors to receive blame for the collapse of the Empire, the eastern Empire’s religion may have been a factor that helped save it. The east was much more homogenously Christian than the west, which still had a lot of Paganism. When Stilicho, a general of the army, was appointed overseer of Honorius, the emperor in the east, Arcadius, was overseen by Rufinus, the praetorian prefect. This position was a holdout from the early Empire, but had become purely civil, the Praetorian Guard having been disbanded long ago by Constantine. Thus Rufinus naturally had less ability to call upon soldiers to do his bidding – in fact, eventually Rufinus would be killed by some soldiers he had screwed over. The successors to his position of power would be the eunuch Eutropius (again, not a military-man) and Arcadius’ wife, Aelia Eudoxia. Eutropius would not last long before also losing popularity, while Eudoxia was actually able to use her Christian piety to establish herself as a very powerful figure in Constantinople. She was able to direct affairs quiet effectively and protect her husband and the eastern Empire. The eastern Empire still had to manoeuvre against barbarian mercenary groups, for example the Ostrogoth, Gainas. However, unlike Alaric, who was able to sack Rome, Gainas was unable to take Constantinople (the population there actually rose up and slaughtered the Ostrogothic garrison in the city) and he was eventually defeated by Fravitta, another Goth. The story of the eastern emperors and court intrigues is going to be told in an upcoming book by Meaghan McEvoy which should be able to delineate the differences further; I’m not really able to devote the time to it right now, but more imperial woman, such as Aelia Pulcheria, Eudoxia and Arcadius’ daughter, played important roles through their Christian piety.

So in summation, the east had economic and geographic advantages over the west, and was able to more effectively manage the fifth century use of barbarian groups in the army and their differing loyalties by a changed approach to the domination of the emperor.