It worked as long as they were properly integrated into the legionary structure. Auxilia were organized, disciplined, led, and ulimately paid and rewarded like the non-auxilia. The system stopped working because they abandoned it. Due to time pressure, they hired entire war bands and armies as mercenaries led by their kings instead of reorganizing them into Roman-style forces. It didn't help at all that the Romans were prone to cheating them out of their rewards.
They did the hiring part that you mention most probably because it was cheaper (in the context of the then Roman economy, that is).
I also think that the much higher level of taxation inflicted on the (local) Roman population certainly didn’t help, as I feel that by the 3rd-4th century many parts of the Roman hinterland were much less populated than they should have been (had not the higher taxation hindered the Roman demographics, that is). In a way the outside-the-limes “barbarians” had the comparative advantage of not having had to pay those very high taxes, and most probably that was good for their (the “barbarians’”) demographics. And on a large enough time scale demographics is destiny.
Yeah, maybe. Money is not the only issue. Battle-hardened troops simply take a long time to be replaced. The early 5th century saw lots of infighting between eastern and western emperors and ambitious provincial rulers, which predictably depleted the army and emptied the coffers of the empire. The Eastern Empire fared better because it was more affluent and more densely settled than the western half.
The fall of the western empire is more accurately described as a fragmentation, after which live mostly went on like it did before. The true downfall of Roman civilization in Italy happened during the decades-long unsuccessful attempt by the Eastern Empire to recover the peninsula.