The unicorn mount is not a WHM mount. It can be done as a CNJ even before you unlock the WHM job quest. The point of the unicorn was to entice people to level up CNJ so they have a basic understanding of how healing works in this game and to help populate the lower level dungeons.

Actual healer mounts (there would be one for WHM and one for SCH) would require similar achievements to the tank mounts and require running endgame content to obtain.