Multiple misunderstandings here. First, the earth is a big resistor, so
the only function of that ground rod is lightning protection, and it
ought to be at the shack, not at the antenna.
Second, the function of radials is NOT to couple to the resistive earth,
but the SHIELD the antenna's field
That's a good question - especially since it relates to your arid soil
- vs my "average" Midwest soil here in Ohio.
Ive done no pattern calculations.
But based on 15 years of reading and personal observations with my 1/4
wave INV-L ( that has grown ( vertically through a black walnut tree)
from 1