Firstly, love the reference to 'I, Pencil'; it remains one of my favourite poignant essays. Adam Smith did have an astute observation on industry and agriculture. However, the same comparison cannot be extended to manufacturing and services. In fact, it is the inability to trade services (although certain services in today's world can be traded) that hinders specialization through division of labour and enhanced productivity. Further, this is responsible for differences in price levels and thus, the real exchange rate. This answer is to elaborate on why collective prosperity is still a far-fetched idea.
Since goods can be traded, it is possible to disperse manufacturing across several countries and achieve division/ of labour. Consider the iPhone. The chips are made in Taiwan but the final assembly takes place in China thus leading to greater specialization and improved productivity.
Now, unlike manufacturing, services suffer from the Baumol Cost Disease leading to the Baumol effect (see https://www.mercatus.org/students/research/books/why-are-prices-so-damn-high). That owing to lower productivity and higher labour costs, prices in the services sector have seen a constant rise. It says that when productivity rises in certain sectors, it makes the prices in those sectors which haven't seen a productivity growth more expensive.
Why? Opportunity cost.
Think about a string quartet which performs Beethoven's String Quartet No. 14 (and extend the example to sectors such as health and education). Compared to the 1950s, it requires the same number of people to play the Quartet even today but since the opportunity cost of their performance is much higher today, it means that the price of their services will increase.
Why is the opportunity cost of their services much higher? This is because the other sectors (mostly manufacturing) have seen dramatic improvements in their productivity. And as per the marginal productivity theory of labour prices (price of labour = their Marginal Product), their incomes will naturally have increased. This explains the price differential between services and manufacturing in a particular country.
What does this mean for price levels in developed and underdeveloped countries? The Balassa-Samuelson effect thus extends this idea further. Since developed countries have seen dramatic improvements in incomes (in manufacturing owing to productivity rises, and in services owing to the Baumol Effect), it concludes that consumer prices in the developed world will be much higher than that of the underdeveloped/developing world.
Further, the TNT (Tradeable-Non-Tradeable) model tells us that since services cannot be traded, but goods can be, it implies that the real exchange rate will be dependent on the prices of the non-tradeable sector (i.e. services). This also partially explains the differences in the exchange rate between developed and underdeveloped countries.
However, this a rather simplistic take on the entire issue. Firstly, like I mentioned before, owing to technology primarily (think online classes and AI in healthcare), the Baumol effect can be controlled. That relative prices in the services need not surge as much. Second, since certain services (think banking and computing) can be done from any place in the world, it means that the price differential in services will not be as much (though the same services are still cheaper in the underdeveloped world).
These multiple factors when accounted for still leaves us with the conclusion of the TNT model. So, to connect to your larger question: yes, a similar version of I, Pencil can surely be achieved for services if there is free movement of labour (think migration laws are easy) and as barriers to free movement of services are reduced (think technology such as the internet). But [Rodrik's trilemma] tells us that such a day is still far away. I hope this gives you some insight. I leave the final connection of this puzzle to you.