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...
TLDR; people considered themselves "Roman", or something adjacent to it, long after the western empire officially disintegrated, and of course in the east it didn't really die.
Even on the outskirts (e.g. Romano-British) people held onto Roman, Christian, and Latin culture at a very deep level. And even when the Anglo-Saxons invaded they were extremely aware of who and what the Roman ruins etc. were, even if they chose not to settle in them in many cases.