Below is the logic why r-g matters for wealth inequality and is a reasonable thing to look at. In particular this happens in so-called "random growth theories" of wealth inequality or "theories with multiplicative shocks." At the same time you're right there is nothing special that happens when r overtakes g and it's kind of irrelevant whether r>g or r<g.
Consider r and g in turn. It's easy to see why r matters for wealth accumulation: if you get a high return you have high capital income and so, for any given level of consumption, you will have high wealth tomorrow. In your example with a $1M endowment and assuming people consume 5% of their wealth a 10% return will mean next year's wealth is $1,050,000 and a 20% return will mean next year's wealth is $1,150,000.
Next consider g: now the idea is the economy is growing at rate g and therefore people's other income sources are growing at rate g. We therefore de-trend everything by g and end up with r-g ("Detrending" here simply means looking at things RELATIVE to the average). So it does make sense to compare r and g, simply because we're thinking about a growing economy so you want to compare EVERYTHING to g.
Also note that these "multiplicative shock" theories aren't really about rentiers vs workers. They are just about people making investments and either getting lucky or unlucky. Here is a good explanation of these theories http://nautil.us/issue/44/luck/investing-is-more-luck-than-t.... Of course, this is just one theory of wealth accumulation/inequality, and there are others with other predictions where r-g would not be so central.
> the economy is growing at rate g and therefore people's other income sources are growing at rate g
Why does a GDP growth rate of g imply that people's other income sources (wages from labor?) are also growing at rate g?
Wage growth in a country depends on confounding factors like worker productivity, relative bargaining power between workers and owners, outsourcing to other countries, automation, and the mix of manufacturing and service jobs in the economy.
It's very possible for the rate of economic growth to be, say, +3% while rate of change in the share captured by workers is flat or negative.
Yes you're 100% spot on. That other income sources and in particular labor income grow at g is just an assumption in Piketty's theory.
If labor income is growing at a different rate, the "correct" version of the theory should have r-g(labor_income). This would probably give even more extreme because it seems like in many countries the growth rate of labor income is less than the growth rate of the economy, i.e. r-g(labor_income)>r-g(GDP).
If you're interested in the dynamics of the economy and distribution if different variables grow at different rates, see here see e.g. http://www.princeton.edu/~moll/UG-slides.pdf
This is based on a bunch of lecture notes and papers: http://www.princeton.edu/~moll/piketty_notes.pdf, http://piketty.pse.ens.fr/files/Piketty2015JEP.pdf (this one is by Piketty and also there it's about r-g rather than r>g -- see the discussion on p.75 and 76) and https://www.aeaweb.org/articles?id=10.1257/jep.29.1.29
Consider r and g in turn. It's easy to see why r matters for wealth accumulation: if you get a high return you have high capital income and so, for any given level of consumption, you will have high wealth tomorrow. In your example with a $1M endowment and assuming people consume 5% of their wealth a 10% return will mean next year's wealth is $1,050,000 and a 20% return will mean next year's wealth is $1,150,000.
Next consider g: now the idea is the economy is growing at rate g and therefore people's other income sources are growing at rate g. We therefore de-trend everything by g and end up with r-g ("Detrending" here simply means looking at things RELATIVE to the average). So it does make sense to compare r and g, simply because we're thinking about a growing economy so you want to compare EVERYTHING to g.
Also note that these "multiplicative shock" theories aren't really about rentiers vs workers. They are just about people making investments and either getting lucky or unlucky. Here is a good explanation of these theories http://nautil.us/issue/44/luck/investing-is-more-luck-than-t.... Of course, this is just one theory of wealth accumulation/inequality, and there are others with other predictions where r-g would not be so central.