I have an inexpensive leather punch and would use that. It is fast and cuts clean holes and is easy to use for the side of a saddle. Placement is not as precise as I might like, so I would work out how to do that.

If I were drilling holes in leather, I would use a "brad point bit" (I have a set for woodworking). These have a central point for locating the hole but cut very clean holes because they cut starting at the perimeter (not the center).