Would people think that it was okay to say that Rome eventually lost, and not try to find some way of saying that the empire survived in some form or another? Or would they try to associate their culture and nation with that of the the Roman empire?
Obviously, there is the Holy Roman Empire, but that doesn't speak for what England thought about the Saxons or Huns. Would they be okay with saying that the Barbarians eventually beat the Roman Empire?
While the Western Roman Empire collapsed in the 5th century, the Eastern Half existed in one form or another until the 15th century.
Regarding the Barbarians defeating the Roman Empire, it would be fair to say that what the Barbarians did and the Franks in particular is adopt and absorb Rome into their political and religious structure and eventually shifted away from the still existing Roman (Byzantine) Empire in Constantinople. The event that best highlights this was the crowing of Charlemagne as Emperor of the Romans on Christmas 800. What this did was de jure reestablish the Western Empire much to the anger of the Byzantines, who both viewed themselves as the true Roman Empire and that no Barbarian had the right to proclaim themselves as a Roman Emperor.
One book I'd recommend looking at regarding late antiquity and the Barbarians' relationship with Rome/Byzantium is Christopher Wickham's Inheritance of Rome. In short, the Byzantine Empire throughout its history called itself the Roman Empire and its people as Romans while the Franks at the least viewed themselves as the continuation of the Western Empire.
Their relationship woth Roman history was complicated, but no - they did not say the barbarians won.
In the Frankish kingdoms, texts like Gregory of Tours' Ten Books say that Clovis, who modern historians consider a barbarian, considered himself to be a Roman consul. Theoderic (king of Ostrogothic Italy) was recognized as a Roman official by the Byzantine government. Anglo-Saxon elites often tried to look Byzantine, and after the conversion to Christianity there was a strong push in england to bring church government and architecture in line with roman models. Visigothic kings in Spain enacted law codes which were based on roman law, and while they distinguished between their roman and gothic subjects, there isn't clear evidence that they saw themselves as breaking from their Roman history.
Most Roman and some Byzantine sources make a big deal about the differences between romans and barbarians, but archaeological evidence, and evidence from the barbarians' descriptions of themselves (when they start to appear, later) isn't clear cut. Some sources make a big deal of the differences between barbarian kings and romans, others stress the roman titles the barbarians held. Some texts play up this tension, as though challenging the audience to decide what qualities really make someone a real Roman (ie, who's more of a 'barbarian' - the barbarians who live in Rome, or Emperor Justinian?).
People in the past were comfortable with complexity, and didn't have the simple understanding of the fall of the Roman empire that you get in popular discussions of the subject today. They were comfortable playing up their roman past when it was useful, and playing up other parts of their history when circumstances called for it.