Your illustration of standing wave is very nice.
The premise is pretty simple: conductors interacting with external EM radiation (that is, radiation emitted from something else), re-radiate the energy (whatever hasn't been damped), but with the wave inverted.
Correct... but
only if the conductor has the same voltage (electric potential) at all points along it. That can occur only if currents are flowing in the wire which create an EM field which exactly cancel the external EM field. This is how a mirror works. This is why a ground plane acts as a reflector, a mirror. This is also why an
infinitely long wire (one far longer than the wavelength) does
not act as a receiving antenna: it would reflect everything and receive
nothing. It would create a standing wave, exactly as you illustrate. How does it do this?
A key fact you've failed to consider: current does
not flow at infinite speed in a conductor, but only at the speed of light. This is true no matter how long the conductor is.
If you accept this reality, then you
can figure out how antennas work. Antennas work as antennas only if they are 1/4 wavelength long (or an odd multiple thereof, as described here
https://en.wikipedia.org/wiki/Dipole_antenna. This is what I attempted to suggest in the previous posting, but you didn't get: antennas resonate in-phase with the EM wave.)
And if you accept this reality, you can also come to understand why an infinitely long wire will
not act as an antenna, but as a reflector. Each 1/4 wavelength along it, at each instant in time, the external EM field
attempts to induce an
opposite voltage, and an
opposite current on it. The is exactly what you state in the quote above. But as time continues, these currents and resulting voltages will continue propagate down the wire in
both directions, and at the speed of light, exactly
cancelling one another at every point along the wire at every instant in time.
A quarter-wave antenna is different,
only because no current can flow in it beyond it's ends. It is as simple as that!
As a consequence, a voltage develops across the antenna (and, 90 degrees out of phase with it) a current flows along it. The ends of an antenna are
current nodes, and voltage
anti-nodes. It is
not out of phase with the external EM field, it is
in phase with it.
Let's stop here. If you understand how quarter-wave antennas work, then you will be able to predict how two joined together to create a half-wave long antenna will work... as a reflector, not as a receiving antenna... and why... current can flow through their joined ends. If this is not yet obvious to you, then you do not yet understand how antennas work. Please go back and try again to do so before proceeding further, as this misunderstanding will only continue to mislead you.
Now, let's turn to linear polarizers. A simple quarter-wave dipole transmitting antenna emits a linearly polarized EM wave. A receiving antenna parallel to it receives it, and one perpendicular to it does not. Same for a second receiving antenna behind the first. No "three polarizer paradox" occurs. The receiving antennas do not alter the polarization of the transmitted EM wave.
A passive receiving antenna does also emit an EM wave, in phase with the one it receives. The two EM waves do interfere, constructively on-axis, and destructively off-axis, together creating a directional antenna. The elements are
not acting as reflectors, but as in-phase re-radiators. But one may
appear to act as a reflector if it is spaced 1/2 wavelength behind the transmitting antenna. (At a receiving antenna, they may
appear out-of-phase depending only on the total difference in path length from the transmitting element to each passive element and then to the receiving antenna. That's why such an array is directional. But most directional antennas have multiple driven elements, with phase differences between them. Lots of examples here
https://en.wikipedia.org/wiki/Category:Radio_frequency_antenna_types)
The fact that all these work is proof your intuition
"At first glance, it would seem to require that the charge distributions on the receiving antenna at a given point in time be the opposite of those which created that point at the original radiator." is incorrect. We observe exactly the opposite. (This fact is so fundamental it even has a name: Lorentz reciprocity
https://en.wikipedia.org/wiki/Reciprocity_(electromagnetism), reflecting its consequences.)
Also sorry you feel math is difficult. You're clearly smart enough to master and use it. Please don't give up on it. It would help you a whole lot. But this is not a math problem, but conceptual one.