Not quite.
The way Britain is governed predates sovereign states as we know them today, and is based on the "realm" concept: all power flows down and is wielded in the name of a superpowerful Crown by people delegated to wield it. So the British Realm expanded outwards, engulfed the island of Ireland, the Isle of Man, the Channel Islands, the island of Great Britain, much of the East coast of North America, the continent of Australia, the islands of New Zealand, and so on. This is why Benjamin Franklin could - accurately - describe himself as British.
Then the realm contracted back and ended up covering bits of the island of Ireland, all of the islands of Great Britain, Isle of Man, Jersey, Gurnsey, and little else. That expansion and contraction took, what, 6-700 years?
Whilst this was going on, the concept of the "sovereign state" took hold: legal entities existing separately from its head of state and with defined borders. The single unified British Realm fractured into individual Dominions which then spun off into sovereign states in their own right. The modern-day UK is the largest remaining self-governing bit under the Realm, but it is not the only one. The other self-governing bits are the Isle of Man, the Bailiwick of Jersey and the Bailiwick of Guernsey.
This explains the OP's original question. The colonies were not integrated into the UK because there was no need (they were governed directly under the Crown) and it was not practicably possible (how do you govern when it takes two weeks for a message to cross the Atlantic?).
Pretty much this. England never bothered with integrating the Isle of Man and the Duchy of Normandy, and with the notable exception of Scotland, Wales, and Ireland, didn't see a need for doing so for the rest of their overseas territories. Heck, having them be self-governing territories would be a much more convenient option for the Crown.