Please can someone tell me where I am going wrong in my calculations.
I am saying that the pitch is irrelevant to the amount of rain a roof
will collect, apart from an exceedingly steep roof and rain that is
almost horizontal.

I don't think you are wrong as such. It is more a matter of
assumptions.

IIRC[1] the assumption made for sizing downpipes is that you have
different downpipes draining different sides of a pitched roof [2]. So
in your 2nd diagramme (the one with the low pitch) a lot more rain hits
the RHS than hits the LHS. If they all drain into the same downpipe it
averages out OK. But if not.....

Another way of looking at it is that rain gauges have horizontal tops.
So all the figures for rainfall are based on that. But it is a fact
that rain don't always fall horizontally. (And the only time I visited
NZ, in Feb 2004, there was certainly some *very* non-vertical rainfall).
So looking at your 1st diagramme, the RHS of the roof collects much more
rain than would hit the horizontal area under just the RHS of the roof.
Less hits the LHS, but again that is assumed to drain into a different
pipe. I think.

If you have downpipes which drain both sides of the roof then it may be
that you don't need to make the adjustment - but I don't know if the
regs. would allow for that.

E&OE of course

[1] a big "if" as I think I met this for A-level applied maths in 1969!
[2] very common in the UK at least
--
R