I agree with this. It’s just too long ago and with too many maybes. As with the butterfly effect, we can be pretty sure that the world would be different, but in what way and how much different it’s impossible to say.
As for the objections raised in the OP. Well some of the obvious differences just in North America would have been the fates of the French Colonies, particularly Lousiana but also the Carribean colonies. Hard to imagine France selling its colonies to Britain. The Spanish-American war and Mexican-American Wars would also not have taken place so no friendly Cuba, drastically different southern US. And the chances of Russia selling Alaska to the English in the middle of The Great Game are approximately nil. Which in turn means the the whole idea of the Western hemisphere being a US sphere of influence changes radically.
And those are just the obvious effects. But if the US remained colony as Canada or Australia did, with the whole world knowing it would send troops and material to any way the moment it was asked, the whole of world history changes radically. If England maintains control, then no WII at the very least since the power blocks would be so very unbalanced that the alliance structures that happened in our timeline wouldn’t occur. Alternatively, the hugely important English foreign territories and constant mutterings of rebellion might have led to England overspending on military and the Empire collapsing 100 years earlier.
It’s just impossible to tell how the world would be different. As Zero-syde said, Butterfly Effect. But we can be pretty sure that it would be radically different. An “independent” US in the style of Canada or Australia is still radically different from the truly independent US. It remains for all intents and purposes a part of Britain until the mid-20th century. It influences the way that Britain spends on its military, the way that it enters into alliances and wars, the way that science advances. The way in which the US is given representation by itself alters history radically. If Americans don’t get a vote, then there’s likely to be constant rumblings of revolution and skirmises, guerilla war etc. And if Americans get some sort of representation, then the politics of not just England but all of Europe alters. After all, the French voted for their politicians in response to the way the perceived English aggression or the lack thereof. So the whole world moves down a totally different path as soon as that issue is decided.
The OP seems to be suggesting that once the rebellion is crushed, the US rapidly gets granted independence, and that the French go on to sell the colony Louisiana and the Russians sell them Alaska despite remaining part of the British Empire. The English monarch and parliament don’t apply any pressure to avoid the Spanish-American war and the Spanish don’t see that as an act of English aggression. England still spends resources exploring the South Pacific despite having vast areas of fertile land and a ready destination for convicts in the American colonies. So it still “discovers” and colonises Australia and New Zealand. The Crimean War played out the same. The Napoleonic Wars play out the same. Germany still ended its alliance with Russia in 1900 because it still doesn’t see any looming threat, so Russia still signs its alliance with England and France, leading to WWI. An independent America still waited until 1917 to send troops to WWI even though all the other Empire nations declared war as soon as Britain did. And so on and so forth.
It’s almost impossible to see any of those events playing out that way in a word where the US is part of the British Empire. And with any of those things changed, the world today becomes totally unrecognisable. Not just cosmetic differences. but major differences.
Imagine, for example, that the Australian continent was settled piecemeal, with a Dutch colony in the North and South West, Russian and French colonies on the east coast and the English restricted to the southern coastline, with New Guinea a German possession. Now imagine the way that change alone is going affect international relations in the 17th century. Imagine a world where, instead of Napoleon rising to power and declaring war on Europe generally, there was instead an alliance of England and Holland against Russia and France due to tensions in the South Pacific. Within a hundred years of the revolution failing, the world becomes geographically, politically, militarily and economically totally different from our own. No Napoleon, no Hitler, Lenin. Unpredictable and weird.
We can speculate endlessly on how the world would be different, but it seems utterly impluasible to suggest it would not *be *different.