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Wow, it's seems that P47's designer just want to make the tip speed approach sonic as soon as posssible. Why?
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4 meter diameter compared to a 3 meter diameter....
4 meters is a big prop and they have to push the tip speeds. Keep in mind, diameter is the most important factor in propeller design.
Good design can compensate, though.
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it says the machs for best efficiencies were in the .7-.9 range.
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They were not dumb at Republic!
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This is probably the reason why P47 was outdived by fw190G from 250 mph(initial diving stage).
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I did not check over your math but your final conclusion is absolutely right.