Understanding Tertiarization of the Economy Trying to understand some of the core mechanics of tertiarization in developed economies.

According to the WorldBank,


  "As incomes continue to rise, people’s needs become less “material”
  and they begin to demand more services—in health, education,
  entertainment, and many other areas. 
  
  Meanwhile, labor productivity in services does not grow as fast as it
  does in agriculture and industry because most service jobs cannot be
  filled by machines. This makes services more expensive relative to
  agricultural and industrial goods, further increasing the share of
  services in GDP."


Trying to understand in a bit more detail some of these claims though, such as:


Why does higher income imply less "material" demands, i.e. products specifically formatted as services rather than goods (couldn't rich people in theory just crave more expensive pharmaceutical drugs, textbooks, and gaming consoles, rather than associated healthcare, education, and entertainment services)? 
Why can't most service jobs be filled by machines?  Does this just mean as technology stands today (vs. roombas, retail kiosks, self-driving cars, etc.)?  What if any inherent characteristics of services make them more difficult to automate?
Why does inefficient production make services more expensive, and this make them a higher percentage of GDP?  (Here I'm just trying to understand the economic principles involved)


Thanks!
Benji
I' m not sure its a law, its what has happened because manufacturing productivity has moved faster. Indeed, the robot economy might well get rid of most service jobs and we'll all be making or programming service robots for each other.

It's related to this paper Buera-Kaboski, Rise of the Service Economy 
But here are some potential explanations:


Maybe after you eat and get dressed you want to be entertained. Entertainment needs novelty, creativity, showmanship, etc. These will always be services because they are a constantly changing stream that needs to be adapted. Imagine a machine that produces new TV shows for you. You would still buy a stream of new ideas for the machine to produce TV shows with contemporary content.
Maybe its best understood as a relative statement. Richer people/countries can have a consumption level that demands more tailoring and specialization that existing manufactured goods provide: Gap produces all kids of clothing, but you want something just a little more specific for you, a tailored suit that's better than the off the shelf one. Naturally, tailored, consumer specific consumption is more a service than a good, because in its nature it is not mass-produced.
In general, goods are produced for the masses, (Bill Gates cannot buy a better iphone than you can), so as a country gets rich, it moves towards services....



  Why does inefficient production make services more expensive, and this make them a higher percentage of GDP? (Here just trying to understand the economic principles involved)


This seems counterintuitive right?: Why would they become a larger part of the economy if they are inefficient? Clearly, this depends on the exact substitutability of products and the rate of satiation.

However, in principle, its not difficult to imagine how this happens. Suppose we can only eat about 2000 kilo-calories a day. Back when we were eating 1000 kc/d, increasing farm productivity increased out consumption of food, and potentially the size of our income we devoted to food until we hit the 2000 kc/d. But as farm productivity increases more, we eat the same, but it costs less and less to produce it, so we devote less and less of or budget to it. Which means we devote more and more to services...