Placing Village Desert Does Not Even Spawn Houses, Villagers, Iron Golem.

Levels defaults to 1, provide a larger value for it to actually generate beyond the village center.

Isn't that setting only available Jor jigsaw Blocks?

No it's an argument in the command as well, and in the structure generation files too.

/place jigsaw <pool> <target> <max_depth> [<position>]

(copied from wiki, not at pc right now to verify is max depth is required or not)

Please provide the exact command used.

/place structure minecraft:village_desert

that means a max_depth of 1, meaning only 1 template is placed; use a higher max depth to create a bigger village.

I was unable to reproduce this in 1.19.2. Please provide a screenshot with your debug screen enabled (F3)