They are the largest military, but they are not number 1 otherwise.
People always look at numbers and think that has something to do with power.
Military power really lies with logistics. You can have the biggest army in the world, but without logistics, you cannot field, move, or supply that army. China is very limited in logistics. Others have pointed out that they have crap for a navy for example. Not only do they not have combat ships, they do not really have transport ships as well. So with its huge army, China can defend itself, but if you do not share a border with them, there is very little they can do to you militarily.
Even if you do share a border with them, their record is not that great. They suprised the UN in Korea, but where then pushed back to the current border. When the ceasefire was signed, the UN had just broken through again.
Following the US withdraw from Vietnam, China attempted to invade Vietnam as a punishment for Vietnam's invasion of Cambodia. It did not go too well for China.
Their military is really built around two things, defence and internal population control and policing.
Also, they are actually down-sizing their military. They are modernizing it still, but the size has been slowly decreasing.
So you are correct in saying their military is overrated. Your reasoning is partially correct.
Where China actually has power is strategically. They have a handful of nukes, not enough to really do anything, but enough to keep off invaders. They also understand information warfare, meaning they can get their idea across better then a lot of countries as well as obtain information. As you already pointed out, they have a lot of economic power as well and they are not afriad to use that economic power to crush a 3rd world nation economy if it poses a local economic threat to one of their goals in Africa for example.
The idea they can buy out the USA is far fetched. If they tried it, they would just sabotage their own economy since thier currency is artificially pegged to the dollar. If they crush the dollar, they crush their currency too. If they tried to un-peg their currency from the dollar, their currency would collapse on its own.
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Paul,
I agree with most of what you said, but even mobilized, they would have a serious problem playing catch up in the naval department. On the land, they could do that, but China does not have the infrastructure to do a rapid build of a navy. Only the South East quarter of China is industrialized. The other 3/4th is much the same as it was 100 years ago.
Also, even if they did build or buy a navy, they have no real core of expertise on how to use a navy.