Bobby, simply check with any surveyor and ask them about how to solve for an altitude of any object if the baseline distance from observer to bottom of object is known.
They will CLEARLY STATE to you they diagram it in the way I PRESENTED, not your method.
It's not the method that's the issue. You just need to make sure you've got the right triangle and thus right ratios.
There's nothing wrong with your opening post calculation, as long as the sighting point that's 3' behind the 10' pole is at the vertex of the triangle giving you the ratio.
If the base of the 10' pole is at the same level as the instrument or eye making the alignment sighting, then you've got the correct triangle and you can use the values of the sides of the triangle using your ratio method. But if you're standing 3' behind the pole but you calculate as if the triangle corner is at your shoes, then you've got the wrong triangle and your ratios using that wrong triangle will give you an incorrect answer. The vertex of the correct triangle will be further along the baseline behind you and the angle to the distant object that you've aligned with your eye from 3' behind the pole will be shallower than the angle coming from your feet.
The problem isn't your method the problem. It's that you said height of the aligning eye doesn't matter. It does. It matters to surveyors.
It's been stimulating, but I can't make it any clearer. If we're still not on the same wavelength, good luck to you. I have to move on.