The Brits did win. America fell back into the British sphere of influence following the French Revolution and the US's humiliating defeat in the War of 1812. America continued to fulfill the role British Aristocrats intended for it, as a subservant trading partner and expansionary military outpost, minus the real cost of empire falling on the backs of British aristocrats. While Americans colonized the rest of the continent, the British Empire continued to expand globally for the next century, hitting its apotheosis in the Crimean War, and then going into retreat following WW1 and WW2.

But even then, British financial interests embedded in the American post-war system continued to enrich themselves enormously straight through Thatcher and Blair. It wasn't until the '08 Financial Crisis that England truly decoupled from the American economy and plunged into irrelevancy.

A British "victory" against American rebels wouldn't have deterred subsequent uprisings, just delayed them for another few years. The bleed on the UK treasury would have continued until America was formally integrated into the British Commonwealth or they were officially cut loose. The American faux-democracy of the 19th century was in the best interests of all parties in the long run.