He was informed that the North was agitated by a furious tempest; that the irruption of the Huns, an unknown and monstrous race of savages, had subverted the power of the Goths; and that the suppliant multitudes of that warlike nation, whose pride was now humbled in the dust, covered a space many miles along the banks of the river. With outstretched arms and pathetic lamentations they loudly deplored their past misfortunes and their present danger; acknowledged that their only hope of safety was in the clemency of the Roman government; and most solemnly protested that, if the gracious liberality of the emperor would permit them to cultivate the waste lands of Thrace, they should ever hold themselves bound, by the strongest obligations of duty and gratitude, to obey the laws and to guard the limits of the republic.
I think it is more than fair to describe this as "taking refugees", though the Romans didn't really have the option of saying "no", because the consequence would have been more immediately ruinous.
It was thought expedient that an accurate account should be taken of their numbers; but the persons who were employed soon desisted, with amazement and dismay, from the prosecution of the endless and impracticable task;(67) and the principal historian of the age most seriously affirms that the prodigious armies of Darius and Xerxes, which had so long been considered as the fables of vain and credulous antiquity, were now justified, in the eyes of mankind, by the evidence of fact and experience. A probable testimony has fixed the number of the Gothic warriors at two hundred thousand men; and if we can venture to add the just proportion of women, of children, and of slaves, the whole mass of people which composed this formidable emigration must have amounted to near a million of persons, of both sexes and of all ages.
http://www.ccel.org/g/gibbon/decline/volume1/chap26.htm
Yes, the Romans did use Goths in their military, but this particular situation was quite a bit different.
The ancient empires weren't modern nation states. People, particularly the Roman designated barbarians were more mobile. The Romans used these peoples as buffer states to keep other tribes migrating from the East away from their borders.
Even the "Roman" lands weren't necessarily Roman. There were petty kingdoms and tribes in there too, with scattered Roman settlements. As the empire declined economically, they started depending more and more on the non-Roman kingdoms to fight as mercenaries, etc.
They played a chess game with the people around them, and the right to settle or not within their rather arbitrarily defined borders was part of the game. Remember that the borders that they gave rights to enter were borders they established through imperial conquest anyways...
The Franks and the Goths both Romanized and created successor states that mimiced the Romans. To many of the people who lived in these areas the transition from Rome to these new states was so gradual they never fully clued into it. The successor states of Francia, Visigothic Spain, Lombardic Italy, etc. all inherited most of the legal and cultural framework of Rome.
Additionally the Western part of the empire fell for reasons probably not much related to barbarian invaders but more to economic and political decline. Taxation models shifted, landholding models shifted, raising large successful armies became harder, and the empire became harder to defend.
And the eastern half of the Empire continued with significant vibrancy for quite some time.
I really like this book on this topic: "The Inheritance of Rome: Illuminating the Dark Ages, 400-1000" https://www.amazon.ca/Inheritance-Rome-Illuminating-Dark-400...