I have to agree with the excellent points allready made, especially about switching hands at turns, theres allready enough rope and prusiks to worry about, without complicating things with an axe leash
My personal experience with leashes has been during beginning mountaineering practice sessions with large groups. The main reason I heard for the leashes, was to keep the leash from becoming a falling object hazard to the rest of the group. Additionally, the slopes where self arrest practice generally occur are not super steep where the leashed axe is going to become flailing object hazard to the climber.
One last point, I use my axe most during summer scrambles where short snow stretces are interspersed with rock and being able to quickly stow the axe behind my pack and access at the next snowfield helps me move quickly.