Reduced centralized administrative capacity due to consecutive crises of succession in the third century conspired with a large wave of Germanic refugees fleeing the huns that Rome would have traditionally repelled or redistributed throughout the empire in such a manner as to diminish their capacity to coordinate political demands or insurgency.
In the end, they got the worst of both worlds. The Romans failed to properly treat, repel, redistribute, or contain the refugees. They also failed to dismantle the leadership structure of the germanic tribesman before they got into roman territory. As a result, the Germanic tribes became indendent armed political blocs within Roman borders capable of threatening and negotiating with roman authorities for concessions, land, and power.
After the Diocletian reforms, the administration was more centralized and powerful than ever. It would absolutely have been possible to incorporate the Germanic tribes under the old auxilia system, and that would have prevented the formation of such "armed political blocs".
But that would have required paying them more or less like regular legionnaires. And Rome didn't want to spend that kind of money anymore; maybe it didn't have that kind of money, due to the economic troubles caused by the crises of the thrid century that you mention.
You make a good point, but here's how I understand it - and I'll cop that may be wrong
Diocletian's reforms did a lot to recentralize the empire, but by the the late 300s during this influx, the nation had been wracked by the wars of the tetrarchy, and a cavalcade of small civil wars instigated by would-be usurpers. The iirc military relied more heavilly on conscription, and the better part of it was deployed in service of the emperor's protection, leaving border territories defended by fewer units more dependent on local governing authorities to provide supplemental irregular forces.
The instability at the upper tiers of Roman government from successive administrations executing purges in various states of paranoia, the vacillations between administrations favoring/disfavouring Christians in different quarter of the empire, and the empire's financial troubles had also done a lot to degrade the Roman bureaucracies institutional knowledge, and capacity to plan/execute policy.
As a result, the roman empire was institutionally incapable of managing an uncontrolled mass migration of the scale that was instigated by hunnic expansion in eastern Europe. This wasn't just a matter of being incapable or too miserly to integrate germanic soldiers into the military, it was a matter of integrating entire nations into Roman society at a scale which Rome had never before experienced in a context far more complicated than an occupation of foreign lands.
Irrespective of their pay, Germanic soldiers weren't effectively granted a place in Roman society because Rome had no capacity to give one to them or their families, and at the same time did nothing to prevent them from staying connected to their tribes or their leaders.
In the end, they got the worst of both worlds. The Romans failed to properly treat, repel, redistribute, or contain the refugees. They also failed to dismantle the leadership structure of the germanic tribesman before they got into roman territory. As a result, the Germanic tribes became indendent armed political blocs within Roman borders capable of threatening and negotiating with roman authorities for concessions, land, and power.