The angle of elevation of the sun is not required, that is correct. But you can use it to find the altitude of the sun in the exact same manner as you have described.
No, you cannot.
Surveyors have been using the method I describe for thousands of years and there is nothing required about the angles of the top of the object(s) being measured.
You have no clue about what you are writing.
Go talk with or write any surveyor and ask them if the method I propose is or is not a legitimate method for measuring the height of objects.
With the 90 degree angle from sun to Earth, and the distance to this point know, the angle of the sun up from the horizon will allow one to find the distance to the sun. Do you dispute this? If not I'd be more than happy to run some numbers, but I'm not going to waste my time if we can't agree on the base principles at work, and I have my doubts at present when you claim your experiment and the one done by Eratosthenes have 'nothing to do' with one another.
Yes, I do.
Surveyors do not use this method to determine the height of objects.
One does not need to know the angle of the hypotenuse.
One need only know the baseline distance from the object being observed to the vertex and the distance of the interceding pole to vertex.
This establishes a precise ratio of right triangles formed without any conjecture.