My eyes widened a little bit at your question. Only because the question itself goes to the heart of a lot of historiography over the fall of Rome. In fact, one could almost trace the evolution of modern historiography, on the basis of how this question is answered.

For example, was the employment of barbarian armies the result of a lack of "martial vigor", military defeat, political necessity, economic convenience, economic requirement, demographic shortfall, etc.

So with that out of the way, I'll express my personal understanding as to the answer of your two questions.

(why) would (the empire) want to take the strategic risk of hiring tens of thousands of soldiers often sharing alternate leadership structures and loyalties

In one simple word, Adrianople.

Barbarians had invaded the empire before, as well as had been settled in the empire before. But those barbarians had been defeated militarily, and resettled in scattered groups, with their previous leadership structures dismantled.

The Goths however, defeated the Roman Army sent to confront them, and were thus roaming free inside the empire for several years. Only through negotiation were they induced to settle, and even then they kept their leadership structure intact. Though the goths would be defeated in other battles with the Roman army, they would never be defeated so decisively as to force the dismantling of their hierarchy, and by that point, with the Roman economy crippled by the Vandal seizure of its tax base of North Africa (which eliminated the ability to pay for a lot of Western Rome's standing army), the Goths were more useful to the Romans as a source of military manpower, despite their constant agitation and usurpation of territory.

did contemporaries see it as a risk or merely another standard aspect of Roman military policy?

Bit of both. There were constant Roman distinctions made to prevent barbarians from becoming full Roman citizens, or becoming emperor. The "fifth column" fear was also a key reason why there was a slaughter of germanic troops after the death of the barbarian general Stilicho. However, by the mid 5th century, their role was indispensible, given the downward spiral of the Roman economy, as is exemplified by the fact that Aetius (himself a barbarian general) had to cobble together a Roman army from a whole host of barbarian nations in order to fight off Atilla at the Battle of Chalons/Catalaunian Fields. There simply weren't enough Roman troops available to throw into battle. And by this point, the empire certainly wasn't maintaining its formerly extensive line of limitanei soldiers on the frontier. The barbarian nations inside the former western empire were simply too many.

So to go back to your main question, how significant was the employment of barbarian soldiers into the Roman army a factor of Western Rome's problems in the 5th century?

It was certainly a factor, but not a big one at the beginning. The employment of barbarians was not the problem. The inability of the Roman army to employ anything BUT barbarians was. But that's more a side effect of a multitude of other things going wrong with the empire at the same time. Although one could certainly argue that all of those stemmed from barbarian invasion.

Which means barbarians themselves are the problem, not the barbarian employment in the Roman army.

According to at least, one historiographical view. There are of course alternate ones. =)

To supplement bitparity's answer a bit, specifically in relation to your second part 'how significant was it a factor in the Western Roman Empire's problems...' - I'll try and outline some of the historiography:

To see the incorporation of barbarians into the army as a major problem normally follows the idea that the Gothic War of 376-382 and Adrianople was the start of the fall of the Roman Empire - shattering the ability of the Roman Empire to effectively subsume barbarians and allowing the growth of separate power structures within the empire. That can be traced back as far Ammianus Marcellinus, writing in the late 4th century, who called them the 'future destroyers of the Roman state'. That then places Alaric's troops and their rebellion and sack of Rome as a direct result of that failed incorporation into the Empire and sees the Empire into inevitable decline from there. Clifford Ando for examples identifies this as the key point and the Gothic political organization as unprecedented for a barbarian group within the empire.

Other historians have disagreed - Walter Goffart for example has taken a rather extreme view in which he argues that barbarian incorporation into Roman armies was necessary and that even later barbarian migrations were not harmful to the empire. This is basically positing an entire shift from the 'Romans/barbarians' division and saying that actually the two were much more integrated than that - in which case your original 'why take the risk' question doesn't make much sense - it wasn't a risk at all. From that view Adrianople was only a result of local commander's incompetence, subsequently followed by a decade of service to the Empire and even post-Alaric's rebellion, an eventual withdrawal and a 'normalisation' of relationships. For a contemporary view that supports that, we can see the early 5th century Christian writer Orosius who viewed Alaric's successor (Aathulf) as a loyal soldier of the Roman Empire using the Gothic forces to 'restore' the Empire.

An in-between view might be Chris Wickham's, who argues that the Gothic incursions were serious but that by about AD 425 (following the abandonment of Britain, Spain and parts of North Africa due to Vandal or other barbarian migration) the Empire had stabilised. Only a change in identification by local elites in the second half of the 5th century AD was the problem - more a result of endemic imperial weakness than the specific incorporation of barbarian soldiers.

That's the historiography - I'd suggest that the only way you can see it as hugely significant is if you feel that a) Alaric's forces crucially weakened the empire by allowing a simultaneous barbarian incursion which resulted in the eventual loss of North Africa and that b) that weakening was only possible because Alaric's Goths were incorporated as a separate force. But even then, take another step back, and that was only the case because Roman military strength had deteriorated beforehand.

For me personally it's hard to get away from the conclusion that the Roman Empire never really fully recovered from Adrianople, but the incorporation of barbarian soldiers was only one aspect of the subsequent problems of the Empire. 'Alternate leadership structures and loyalties' sprung up all over the Empire, not just amongst barbarian units within the Empire and arguably more integrated Roman usurpers had a more key role in the Empire's problems.