It's not that it isn't possible, it's that given the resources we have, it isn't optimal.
Your thesis is wrong not because it can't be done, but because the competitive gains (transportation, communication) don't outweigh the gains to doing much higher value-added tasks. (Not to speak of the potential costs in having to pay higher wages, deal with environmental concerns, etc, but i'll leave those out because they're political, not fundamental.)
Let's say the US only has 10 engineers, but they're really really good. China has 50 engineers, but they're less good. (Good being a stupid shortcut to saying they have more domain expertise/resources, not Jingoism about the quality of Chinese engineers).
Now the US engineers _could_ start a factory making iPods, and they'd be just as good as the Chinese, or just a little better and sell for around the same price, with a profit $X. It's relatively low startup cost, and has relatively low barriers to entry.
Or, the US engineers could start a factory making specialized composite-material airframes for space ships, which the Chinese lack the ability to build at all (so far), with a profit of $50 * X. It's much higher startup cost, and has huge barriers to entry, so there won't be competition for years.
The US engineers can build the wings, buy iPods from the Chinese engineers, and still pocket a huge difference! Meanwhile the Chinese engineers are thrilled to build ipods, because it's the first step in learning the skills to build Rockets. (Nonsense example, but you get the idea) This is basic comparative advantage with respect to a value-added chain.
Your theory would override this IF we had unlimited engineers and a basically level playing field for production costs at the low end, but we don't for tons of reasons, mostly around regulation (which I'm not saying is a blanket bad thing).
With only so much capital, why make iPods when you could be making satellites?
Now of course this example is trivial for lots of reasons, but the general pattern has been playing out for 100 years.
As nations gain expertise and working capital, they move up the value chain and outsource what they lack the resources or environment to build cheaply. Historically this gain in ability has been accompanied by an associated gain in wages/worker protections (for lots of complex reasons), which even further propels you a nation to move up the value chain, since you only want to build the most profitable possible things that you can.