The beginning of your response is saying that the EIRP is the same. You are correct. Please read the last paragraph of my previous post.
Your math is all correct, including FSPL calculations. However, FSPL is about loss in free space irregardless of what antenna or power output is used.
Note I am referring to someone standing in the line of maximum power direction, I realize the shape of the coverage area will differ with antenna type. However, in the path of strongest signal, I don't see any way that the same EIRP will propagate differently because of different antennas. The reason for this is the antenna gain is accounted for BEFORE we arrive at our final EIRP quantity.
We are in agreement that we are not increasing total power. However, the concept that you are currently missing is that an antenna concentrates that power. The higher gain the antenna, the smaller area that receives the power.
Again, I go the the magnifying glass. Let's say the exact same amount of light goes through a 4" piece of flat glass and a 4" magnifying glass. Which allows more light (power)? Neither, they are equal. Which one concentrates the light? The magnifying glass. More power is concentrated on the same point.
You say we get a better signal if we use a higher gain antenna, and I agree, however, if we reduce the power, then it cancels out in the advantage. 100mwEIRP is 100mw EIRP.... It's the same...
This is not correct. Given the same EIRP, the higher gain antenna will have more signal in the direction deemed by the antenna design. A parabolic grid will concentrate it into a stream, an omni-directional into the flattened dough concept.
It's not more power, it the same power concentrated. The reverse also exists. There is less power everywhere else.