Note that the Ultrastable Oscillator they use for their 30 MHz frequency reference has a short term stability of 1 part in 10 trillion! To compare, TCXOs used in commercial circuitry often have a stability of around 1 part per million.
While impressive it's not clear why such a low phase noise oscillator is needed. This oscillator looks like it takes up lots of space and power. So some justification for the device should be presented. Furthermore its possible to create a robust FEC that would be tolerant of burst errors due to phase noise. So the paper is incomplete without some mention of phase noise and the FEC used.
Other than the high end parts used in this device, it is fairly low-tech compared to most modern communications systems. That is a good thing for something like this I guess.
Keep in mind this was designed in 2002 using technology that was considered rock-solid & mature at that time. Spacecraft technology always seems outdated because of the lead times required: on something like New Horizons, it's between 15 and 20 years from component design to being used for its primary mission.
In addition, as another commenter noted, the reference oscillator is used for REX (the radiometry experiment on board New Horizons) and quality of gathered measurements for this experiment was one of the primary needs here: being able to use the component for the communications subsystem is just good power/mass engineering at work.
I believe the main reason is for navigation. They need a stable frequency reference because they use its delta, due to Doppler shifting, as the measure of the spacecraft's radial speed relative to earth.
From the paper, even this amazing stability isn't good enough for their navigation needs, by itself:
7. NAVIGATION SUPPORT
The downlink frequency from the New Horizons transceiver is
derived from the USO, as discussed previously, but even this
excellent frequency reference is not sufficient to support
precise and accurate Doppler velocity measurements at the
target level of 0.1 mm/s throughout the mission. In order to
remove the impact of frequency drift and bias in the
reference oscillator, New Horizons has implemented the
noncoherent Doppler velocity measurement approach [2] which
was successfully demonstrated in 2002 during the demanding
Earth orbit phase of the CONTOUR mission [3].
