You are missing one important factor in your analysis (which took me a long time to figure out). People with lower income spend more on consumption goods, than the higher income individuals, who spend more on investments. This is important, because the rich are investing much of their wealth in growth, whereas the poor are spending on necessities and happiness. Transferring money from the rich to the poor also transfers money from investment to consumption, which has important effects on long term economic growth.
One must remember that the US grew only 0.5% faster than the UK through the 20th century, and this led to a large difference in outcome.
There's one important question you've forgotten to ask: what exactly are the wealthy investing in the growth of? The answer, of course, is the production and sale of consumption goods, either directly or indirectly - all of their investment returns have to come from there eventually. Take that away and you'd just be left with a bunch of investors trying to make money from selling other investors ways to make money from yet more investors, in a mass of ponzis and swindles producing no real economic output.
One must remember that the US grew only 0.5% faster than the UK through the 20th century, and this led to a large difference in outcome.