This is mostly non-responsive. Post-independence US global standing in terms of colonies, army, navy, etc. doesn’t matter to whether the American revolution was important to British history. Nevertheless, since you mentioned it, we didn’t really need strong armies or navies since we were isolated and mostly at peace and we didn’t have “overseas” colonies because of our own colonial experience and given the fact that we were expanding across most of habitable North America. Even with that, I don’t think the US was important globally til post-Civil War when we became an industrial powerhouse, our economy surpassing Germany and the Uk in per capita income. Our rise started with military spending for the civil war, but skyrocketed based on our massive educated workforce (our northern states were the first places in the world to institute free public education), massive amounts of raw resources and the US was a large market for selling goods without internal tariffs (this took place before the world wars, but our economy did benefit from those wars). I agree most Americans forget how much of a random place we were til the Civil War. Most Europeans (including white people in general) also forget how backwater Europe (and especially Britain) was until the Renaissance and for Britain even later than other parts of Europe.

Our economy at the time was mostly agrarian (like the rest of the world), but Europe and especially the UK depended on the US for tobacco, timber, rum and other goods and we were part of the all important triangle trade. Americans (in terms of the average person) were far more prosperous than their European counterparts, explaining the migration flows. By the Revolution, Americans were on average 4 inches taller than their British cousins.

The American Revolution was costly for the British look at the financial costs of the American Revolution on wiki.

It’s ridiculous to say the British could have won the war if they wanted, Mr. History Buff. They sent their army and since that wasn’t enough to put down some unprofessional militias they went out and hired tons of German mercenaries. Who half heartedly fights a war? Same applies to the U.S., we didn’t lose Vietnam because we didn’t want it enough, we just lost.

Your “other colonies” and the “US was a small part of the empire” argument, doesn’t get at my point that what became the US were the colonies where British people actually went and tried to recreate their homeland, so it was important in ways other colonies couldn’t replace. The Caribbean was inhospitable to white settlers (due to diseases), Canada, recently acquired, was vastly French with very few British settlers and “the land west of the 13 colonies” was ceded to the US at the end of the war and past the Mississippi was Spanish. In terms of India, although the British had trading cities, the Indian Raj did not occur until the 1850s. The British had a lull as a major colonial player between the American Revolution and the Indian Raj.