Why would they need a new emperor when there was very much still a Roman emperor in Constantinople?

The Roman Empire did not end in 476, arguably it didn't end until 1204, 1453, or even 1922 if you want to get extremely pedantic.

The Eastern Roman Empire, or Byzantine if you prefer, was still very much alive and well following the "end" of the Roman Empire in the west. Various barbarian rulers still had to acknowledge the supremacy of the emperor in Constantinople. The emperors for their part were happy to meddle in the affairs of the kingdoms cropping up in the western Mediterranean. This influence could take several forms, such as paying off certain realms to fight with each other (the Ostrogothic invasion of Italy was attributed to Roman influence) and later outright conquest under Belisarius in Africa, Italy, and parts of Iberia. So yes, there was a Roman 5th column, they just happened to often be a column of Roman infantry spreading Roman rule over former Roman territory...

Moreover, the barbarians now living in the former empire went to great lengths to portray both themselves and their rule as consistent with Roman norms and times. This idea of Romanitas or imitating or embodying the ideas of the Roman elite was still popular in large parts of the former empire. The senate continued to meet, roman baths were still in operation, Latin philosophy continued to exist, and the Roman Church was downright thriving, public offices were handed out, and so on down the chain of "roman" practices.

However in certain parts of the former empire it was quite evident that things had changed, and this became clearer over time, but was not immediately apparent in the aftermath of Roman collapse in the western Mediterranean.

Britain was overrun by newcomers setting up their own polities that derived their legitimacy from continental antecedents and invented legends of conquest and divine lineage. Gaul, Iberia, Italy, and Africa all had new ruling elites who attempted to introduce their own customs such as their own legal codes and religious beliefs (usually Arian Christianity) over the Roman population, and as time went on in these realms the facade of imperial continuity started to crack. This was due to many reasons but perhaps most importantly was the collapse in Roman tax systems. Maintaining the infrastructure, both physical and metaphorical, of the empire was dependent upon an enormous tax burden on the population and once the centralize Roman state in the west was fragmented and dissolved it became impossible to maintain the Roman way of life in many urban centers (ignoring the urban centers that were in places like Britain that simply ceased to function entirely as population centers).

However for many people, at least at first, life improved, long distance trade actually ticked up slightly following the collapse of Roman authority in the 5th century. Long distance trade, indicated by pottery from Africa being found across the Roman world, continued for some time after the collapse of Roman authority in Africa, turning up in Gaul, Illyria, and Britain throughout this time. However before the Arab conquests in the 700's it seems that economic activity had receded in almost every area of the Mediterranean world.