China's a bit of a different case - but for contrast, compare it to the USSR. Despite three decades of war, civil war, near-genocidal purges, and some more war, by the 60s and 70s, it has lifted millions of people out of a similar level of intractable poverty. Again, not on the back of its economic system, but on the back of industrialization.
It's hard to stay poor (Compared to world poverty in the 1800s) when you have running water, indoor plumbing, and electricity.
[1] https://tradingeconomics.com/india/gdp-per-capita
[2] https://www.slideshare.net/ashishverma061/growth-of-electric... - Slide 16.
You are extremely confused about the history here. The British left India in 1947; WW2 ended in 1945. India and South Korea had a similar GDP/capita in 1950; they were on opposite ends of the spectrum 50 years later after 50 years of a wide GDP growth gap (during none of which time did British India exist).
I believe more than most people in the path-dependency of politics, especially with respect to things like colonialism. A lot of the sclerotic, Soviet-influenced, bureaucratic strangling of the Indian economy during this period is at least partially attributed to (reasonable) democratic political backlash against the abusiveness of the British economic system, in the form of eg labor laws that have been harming the economy for half a century.
That was James Watt's steam engine, which was efficient at turning combustion into mechanical motion, and it was happenstance that it was invented in 1780, and not 1580, or 1980. People didn't suddenly develop a need to move heavy things in 1780 - it came around then because of advances in steelwork, precision manufacturing, and general refinement of the idea (Such engines were around for over a century, but were not compact, or efficient enough to be very useful.)
All of a sudden, the labour of a coal miner was enough to feed a 10, then a 50, then a 100 horse-power engine. It was the AGI of its era.
http://www.his.com/~pshapiro/steam.engines.html
Why was there a coal mine? Why not use wood or charcoal? Why didn't the Romans go down this path? They invented a toy steam engine.
These things have many causes. Obviously watson's invention was a big inflection point.
But, as for the question "were liberal markets necessary?" my suspicion is yes. But, I don't actually know. I think the answer begins before James Watt.
Why the steam engine did not succeed because Watt, was because the same reason that tri-deck galleons and gunpowder did not exist in the bronze age. Non-trivial practical inventions require a massive pyramid of other non-trivial practical inventions, each of which requires a pyramid of other non-trivial inventions, all of which have to solve a contemporary problem - cheaply then their alternatives.
If liberal markets were necessary for the invention of the steam engine, why was it invented in England, and not in Renaissance Italy? It had liberal markets, access to capital, and was a center of technological and cultural progress for hundreds of years.
There's no particularly satisfying answer to that, other then 'happenstance.'
It's almost tautological to say that energy is behind growth, but it's also independent of economic structure, which effectively boils down to "the manner in which we convert energy to utility".
PS: The China boom looks 'good' mostly due to how terrible the country was run prior to that. Expand the time scale and compare to other countries and things don't look nearly as good.
I don't know a ton about the economic history of these two countries, but prima facie this seems to belie your claim that it was simply energy availability.