Haven't looked at the site to see exactly what it says, given how wrong it usually is on basically everything. But:
The method is not exclusive to points at different latitudes.
It is quite general, you can use any 2 locations.
What you need to know is the distance between them, along the surface, and the angle of elevation of the sun for each point.
Using that, along with the fact that the sun is very far away, you can calculate the radius/circumference/diameter of a spherical Earth.
Alternatively, using the false assumption that Earth is flat, you can determine the height of the sun.
You can also use a method where you only look at the angle of elevation at solar noon. This takes away the error in timing. But this does require the points to be at different latitudes, and you need to know the distance between those latitudes rather than the points themselves.
But regardless of if it goes through the north pole or not, that can't give you the diameter of a flat Earth.
There is no mathematical connection.