"The high-gain antenna has a beamwidth of 0.5° for X-band, and 2.3° for S-band."
At 130-150 AU, the earth is always within about 0.4° of the sun. Since commands are sent on S-band, pointing directly at the sun gets a pretty good signal.
Being drowned out is harder than you might think. The maximum data rate of a weak signal is 1.4 x [bandwidth] x [signal-to-noise ratio]. If you transmit across a 200MHz band, and your signal is a million times weaker than the noise, you can do hundreds of bits per second.