Its complicated

The various Norse settlements and colonies that were established across the North Sea, Russia, and even into North America were all interlinked by trade and some commonalities in cultural practices. For example, ship burials are attested in Rus territory, Sweden, and Denmark, and many scholars attribute the settlement of Vinland as a part of a resource network to bring timber for ships and construction to Iceland and Greenland.

However these new settlements were not directly ruled from Scandinavia, but they were quite connected to them.

I'm going to take Iceland as a bit of a case study since I recently read an article and book on the topic. Iceland was originally settled by the Norse in the waning days of the the 9th century, but the exact date isn't clear.

Once established here, the Icelandic populace adopted their own method of rule that has been termed the Commonwealth. Importantly the Commonwealth was not held in absentia or in the name of any Scandinavia mainland ruler or power. it was its own independent polity, separate from other entities. However this did change in the 1200's. Due to the endemic conflicts between the Icelandic rulers, the members of the Commonwealth decided to offer subservience to the Norwegian throne in exchange for protections from their Icelandic rivals.

Now the type of government in Iceland was unique, but their experience was not. Several Norse ruled kingdoms and settlements retained their independence or they were absorbed into other polities. Greenland and Vinland were abandoned and that is another story.

Let's examine the Kievan Rus a little closer to see an alternate outcome. Peter Heather argues that the Norse settlement of Russia was originally economic in aims, seeking to establish trading posts and market towns to facilitate trade from Scandinavia and Finalnd to the Islamic and Byzantine world.

However kingdoms of the Rus eventually cropped up in these territories. It is a matter of debate how large the influx of settlers was to these areas and whether these kingdoms were formed over existing polities or not. However despite the settlers coming in from Scandinavia, there does not seem to have ever been direct rule from Sweden over these areas in Russia.

These kingdoms such as Kiev, eventually became drawn much more into the Byzantine sphere of influence, with the Kievan Rus accepting orthodox Christianity, forming alliances with the Byzantines, and so on. While they may have been founded by Norse settlers and traders, they were never really a part of Scandinavian powers.

Then there's the issue of the British Isles. I'm not very familiar with the Norse in Ireland or the Kingdom of the Isles, but I am aware that they were eventually subsumed into native polities. The Kingdom of the Isles also experienced brief periods of direct Norwegian control, but was eventually brought into the Kingdom of Scotland.

You bring up York, and this is an interesting example. The Danelaw in England was relatively short lived compared to these other settlements and colonies (not counting Greenland and Vinland). Several sagas indicate that the Danelaw settlements were continually involved with Scandinavian affairs, Erik Blood-Axe is supposed to have come there to rule from Norway almost a century after the original Danish conquests. I don't think I need to rehash all of my issues with the Sagas as historical sources, but here they are at least supported by the historical record.

Following the collapse of the Danelaw, there remained many notable Danish rulers in England who became a part of the English aristocracy. During the attacks on England by Svein and Knut, other Danes were once again able to find positions of power in England. The Norman conquest is finally what divorced England from Scandinavian affairs, despite the best efforts of Norwegian and Danish monarchs.

There was no one experience for the Norse settlements and colonies that can be applied to all of them. The settlements in Kiev eventually coalesced into kingdoms. Vinland and Greenland were abandoned, Iceland, the Kingdom of the Isles, and the Danelaw were all eventually subsumed into other polities, both Scandinavian and British. When they were first founded though, they were not subject to Scandinavian ruler or domination.

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Sources and further reading:

I mentioned Peter Heather's Empires and Barbarians

Gunnar Karlsson's History of Iceland

Sverrir Jakobsson "The Process of State Formation in Medieval Iceland"

I also briefly touched on the sagas, looked over Ibn Fadlan's account of the Rus, and Saxo Grammaticus's Gesta Danorum