I think the first thing is to get rid of the idea that the British Empire was this all powerful thing. Yes certainly a century or so LATER it became a global superpower but around the time we are talking about? Not so much.

From a geo-political point of view, England/Britain had emerged as a small but punchy power in the Elizabethan era into one of a half-dozen European powers in the Stuart Age. While the New Mode Army was rightfully feared, it was only after the Glorious Revolution that Britain truly began to show it had some genuine financial and naval power.

The victory in the Seven Years War was truly where Britain emerged as a global superpower. A new one. And even then it’s reach was limited by it being a sea based one (Britain was only rarely able to compete with the great European land powers in the field; its strength lay in its fleet).

This explains why so many nations were willing to jump into the War of Independence- not that they cared too much about the colonists. More they sought a chance to cut the new upstarts down a size.

To be more precise about your question- first when asking ‘how did Britain lose control of the American colonies’ I find it easier to simply ask ‘WHEN did Britain have control of the American colonies’? Direct rule was next to impossible and it could be argued that the colonies were British only for as long as they wished to be.

Secondly, the explanation as to how the British lost is always simply done. The American colonists were fighting for freedom against British militia and soldiers. The British were fighting the colonists, the French, The Dutch, the Spanish Empire and the Emirate of Mysore. All at the same time.

Thirdly, from a British point of view the American colonies were NOT the prize possession. Britain was ‘Queen of the Indies’ and valued her Caribbean colonies much higher. It is worth noting the way the British prosecuted the war against the Dutch in the Antilles compared to how they did against the colonists.

Finally from a British point of view, there was wide spread support for the colonists (which diminished somewhat after all those other nations got involved) plus serious domestic issues which dominated the politics agenda).

When you consider that the biggest battle during the entire war was in Europe (the joint Spanish/French attack upon Gibraltar) and that while fighting had started in the Colonies it was now leading to a conflict being fought on three continents, enforcing an unpopular war upon people who many sympathised with, stretching the ability of the nation to cope with such a large scale military/political crisis, the final recognition of American independence which humiliating for the king, was greeted with relief by most of her citizenry.

Besides, back in India the Honourable East India Company was starting to chop up and take large parts of India and this, combined with the territory they had taken from the Dutch soon turned their eyes eastward...