The overwhelming loss in this calculation is from the antenna’s radiated energy spreading out over a larger and larger area (despite the directional “gain” factor). I’m wondering: would a probe launched today instead employ a laser to communicate? This would seem to offer many orders of magnitude improvement in the directionality of the signal.
The main challenge is the earth to probe comms for distant probes, since the earth is often very close (in an angular sense) to the sun from the probes perspective, and the sun gives out a lot of black body radiation.
However, due to the shape of the black body radiation curve, the sun gives out relatively less microwave radiation than it does visible light, which might outweigh the advantages of more directionality given by using a laser.