Proponents of a flat tax usually underestimate how unequal incomes are. Any reasonable flat tax rate would result in an increased tax burden for mid to high earners and a substantial tax cut for very high earners.
This paper for the ACCA for example runs through an example implementation. The result is
reduced taxes for those earning less than £22k and more than £75k, and a tax
rise for earners in the middle (pp 47). How is that better than the current situation exactly?
I'd rather see a smoother increase up to a higher top rate, maybe 65% (as suggested by Tony Atkinson), but each to their own.