I'm most familiar with Post-Roman Britain so I can hopefully address this part of your question!

So, as you've said, Germanic migrants (who I'll say are a wide range of different peoples - Angles, Saxons, Danes - as you've said, but also Franks, Frisians, Jutes, and Geats to name a few) did seem to dominate over the former Roman province of Britannia, but mainly the area now known as England.

I'd like to note that this period of history is fairly murky so this is largely based on archaeological sources.

To address the power vacuum; in Britain, this is extremely prominent as essentially the Roman state ceases to function and all the trappings of Roman life which were supported by this also cease to exist. No more towns for administration or villas for economic production.

As Germanic peoples arrived in Britain they were finding a land which for the most part was still in the process of adjusting to the disappearance of the Roman state. As far as we know, there were no cohesive political units which would be able to form an effective response to wide spread migration. There certainly could have been, but we don't know of any.

Gildas, our closest historical source who was writing in the sixth century, describes Germanic settlers contracted as mercenaries to defend against raiders - who turned on the Britons and made war against them, taking much land. This is spotty at best so should not be taken as absolute historic fact, but it does correlate with the dramatic shift in material culture in now Anglo-Saxon areas of England.

In the west of Britain (Wales, Cornwall, Devon and the general southwest) it is possible to identify possible political units forming around newly occupied hillforts and defended enclosures. Which aligns with these areas remaining under British control for longer periods of time.

In the east of Britain it is a different situation. Archaeologically the influx of material culture from Germanic peoples is very visible, and the absence of Romano-British material doubly so. This suggests that people were no longer functioning as they had during the Roman period - Production shifts from providing an economic surplus to subsistence before a new material culture is introduced overtaking any remaining Romano-British material.

So to summarise; the situation in Britain is complex but can be simply described as disorganised at best, and chaos at worst - allowing Germanic peoples to migrate, settle, and in some areas gain complete political control rather quickly - Kent is a key example, where within 25-50 years after the end of Roman control there were people settling in the remains of the former principle city, Canterbury, with new structural types and material culture.

To address organisation, it is fair to say that you may have underestimated the organisation of the Germanic peoples, but also the picture of Germanic peoples migrating then immediately setting up new Kingdoms is slightly simplistic.

From what we can tell, the new populations in Britain were functioning as household units, so a principle family and their household which may have been enough people to form a village sized group of people (around 20-50 individuals is fair to say).

These groups would also ally with neighbouring groups to form alliances through marriage and the exchange of gifts or labour. Quite rapidly an overarching hierarchy would form with some household units more prominent than others eventually leading to the formation of small kingdoms or political units over time, which would gradually swallow their neighbours and rivals.

They were organised on the local level and would frequently co-operate with one another to get things done, but they wouldn't be called Kingdoms until the end of the fifth century and into the sixth. But what they were moving into certainly allowed them the freedom to act in this way, with no formal state structures or armies which the inhabitants had been used to, any response to stop people moving into Britain would have been largely ineffective.

The people that were arriving in Britain were by no means disorganised, but were rather organised much differently to the Roman state, and were positioned perfectly to gain control over areas which were now experiencing a power vacuum or at least were experiencing a lack of control.

Unfortunately I'm not in a position to describe the situation in Spain, France, or Italy but I hope this has provided at least some insight into the situation in Britain.

Sources:

Fleming, R., (2011) Britain After Rome: The Fall and Rise from 400-1070. London: Penguin Books

Laycock, S., (2008) Britannia - the Failed State: Tribal Conflicts and the End of Roman Britain. Stroud: The History Press.

Carver, M., (2019) Formative Britain: An Archaeology of Britain from the Fifth to Eleventh Century AD. Abingdon: Routledge