Drizzle is a trifecta with WX radar.
First the extremely small particles tend to be stealth, and have much much less reflectivity than larger droplets or snow.
Second, drizzle tends to be low to the ground, and to the radar processor, may look like the ground to anti-clutter algorithms. The low to the ground can also impact how much energy is radiated there, also an anti-ground clutter measure.
Third, drizzle by it's nature tends to be rather stationary. The air it is in is normally stable, and there is little movement for a Doppler processor to pull out. The radar processor handles echo data, not just Doppler data, but the Doppler data is used to enhance the sensitivity of the unit. Generally it works well for moisture which is moving, but drizzle has little motion. Except downward, which is orthogonal to the radar, and therefore not a factor.
Similarly, most clouds, and smoke may not show up on WX radar. Birds may, as their radar cross section (RCS) is larger, then tend to move, and they tend to move as a mass of points, all which make them more easily trackable with radar.
In summary, the reasons why Doppler WX radar tends to not show drizzle well are: 1. Extremely small particles, 2. drizzle is low to the ground, and 3.drizzle is stationary and therefore not enhanced by Doppler processors.