The bug
The drowned mob when used in a spawner appears to only spawn in its natural water biomes (e.g. river), in or out of water, but will not spawn in a non natural biome (e.g. desert) under any circumstance.
This contrasts other biome specific mob type such as the stray, which will spawn in any biome as long as they are used in a spawner.
Linked issues
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Comments 8
Drowned spawned just fine for me in water in a desert. They just require a water block below the block where they spawned. create a pool of water around the spawner and they'll spawn. I don't know if they should or not, there seems to be conflicting information on this tracker about that. MC-175110 suggests that its a bug if water mobs spawn outside of water, but MC-250054 suggests its OK if water mobs spawn outside of water. I don't know what the correct answer is.
yes, the drowned will spawn in any biome if you are below y=64, however anywhere in a non-water biome above this the drowned spawner will not work, regardless as to whether there is or isn't water around
@Adam, it looks like height requirements are WAI as suggested here in MC-238526
Yes, thinking more about it now this does appear to be WAI, spawners generally work based on spawn conditions being met and the height/biome rules do seem to be conditions for the Drowned mob. The note in the OP about Strays being able to spawn outside their expected biomes misunderstands the Stray's spawning requirements from what I can gather.
My testing indicates to me that they will work in any biome, but only below sea level.