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I doubt the diagram would be as famous and respected as it is if it had been as flawed as the OP suggests!
A radial representation is a different type of graph entirely.
The text is correct: in these diagrams the area is proportional to the statistic, as in all "polar area" and pie charts.
If there is a weakness, it is that all areas are (as the text says) "measured from the centre"  so the area which represents deaths from wounds is the blue+pink+black, not just the blue bit (as you'd intuitively expect  so there is no "order" to the segments).
It's obvious if you look at the original figures:
For example, in January 1855 deaths from diseases were 2761, from wounds were 83, from all other causes 324.
If the chart was radial, pink+black would only come 1/7 (407/2761) of the way to the edge.
Whereas actually, pink+black should come about 1/3 of the way out (root2761/pi)/(root407/pi). [if my rusty maths is right]