I've been trying to get a drowned chicken jockey to spawn, with little success. I'm aware that a naturally spawned chicken jockey will separate into a chicken and baby drowned, see also MC-209784. As far as I know from the MC gamepedia wiki, this is expected behavior.
So I have been using a second mechanic to get a baby drowned jockey, by having the spawned baby drowned check for a nearby unoccupied chicken. This mechanic has worked for baby zombies and baby zombified piglin. Sadly, as seen in the attached picture, the drowned jockey spawn for this second mechanic seems bugged. There you can see that the baby drowned is standing inside the chicken, rather than mounting it.
Note: for command block-spawned drowned, the naturally spawned drowned jockeys are persistent: they don't separate into a chicken and drowned. Also, command block-spawned baby drowned will easily hop onto a chicken.
Steps to Reproduce:
Make a water column above a river biome, allowing drowned to spawn. Here I used flowing water.
Have a nearby chicken (or several) placed on glass blocks, making sure the glass doesn't touch the water.
In 0.25% of the spawns, a baby drowned will spawn that should attempt to mount a chicken.
Observed Results:
The baby drowned spawns at the location of the chicken, but does not mount it.
Expected Results:
The baby drowned should mount the chicken and become a chicken jockey.
Since the submission of this 'bug', I've figured out a different way to get a drowned chicken jockey. So far, I've only tested the following for naturally spawned chicken jockeys, not for baby drowned that will attempt to mount an available chicken.
Jockey separation
The trick/observation here is that a drowned chicken jockey will not separate when it spawns inside a water-source block that has air bubbles in it. Either soul-sand (rising) or magma (falling) induced air bubbles will work, although in the last case the chicken will eventually die due to fire damage.
Getting and keeping a drowned jockey
To farm/get a drowned chicken jockey, I've made a 10x50x10 water column (all source blocks!) encased in glass, and soul sand on the bottom. After about 30 mins, a first drowned chicken jockey rose to the surface. I've named both the chicken and baby drowned and transported them in a boat. Especially on servers, chicken jockeys may separate randomly; so if you want to keep them: put them in a boat at their final resting place!
Further testing
I have yet to test whether the 'hop on' mechanic of a baby drowned will work when the baby drowned is spawned in a water-source block (in stead of flowing water, the situation of the original bug filing)