John Ferrell wrote:

I will spend more time here. The last model I worked out with a
perfect ground appeared to produce a 0 degree take off angle.
. . .

That is the correct result. The pattern is exactly the same as half the
pattern of a vertical dipole, which has a maximum at zero degree
elevation angle, except that the gain of the vertical is 3 dB greater
because the same power is concentrated in only one hemisphere.

Roy Lewallen, W7EL