Yes, it does make sense to me. I think conclusions drawn from this thought experiment are not entirely correct. Under your experiment when sending you spatially create a beam - there is this beam as wide as parabolic dish and no light anywhere else. With zero interference (using the word quite freely) this beam "will be just as strong no matter how far away the receiver is". With roles reversed, you do not necessarily get directional transmitter. For the sake of argument consider omnidirectional light bulb (the same bulb without parabolic dish) as sender. Classical intro/ELI5 description of transmit power is fixed number of spatially evenly distributed omnidirectional "rays". The larger the distance, the lower number of "rays" you receive per unit of area (receive power scales with surface area of transmit profile, usually r^2). By adding a parabolic dish you effectively increase receive area to size of dish instead of sensor size. When sending, the area over which power is dissipated is constant instead of scaling with distance.
> You're claiming it won't matter how far away the sender is from the focus, which makes no sense.
You are correct in that it makes no sense, but the claim was "every improvement in sending is completely equivalent in receiving for a system that uses the same antennas for both directions", key word improvement. While this is not entirely correct, this approximation (that antenna improvement works both ways equally) is pretty close to reality in practical far field applications.