The charging system on most small engines is by magneto system. The flywheel has magnet in the edge which induces current to flow in a coil (pick-up)mounted very near. The coil has one side connected to ground and the other to the battery. The mowers headlights by the way are powered directly from the magneto. That is why they only work if the engine is running.
Make sure wires are not disconnected from pick-up coil to battery. Note: Probably will only have one wire as the frame serves as ground. Also the wire probably won't go directly back to battery, but to a common point like the starter solenoid or key switch.
As for hard starting, probably not battery or starter if mower is old, exhaust valve clearance (lash) too slack. Causes compression not to be released quick enough because the cam moves but the valve does not open immediately and the starter can not overcome the compression on big one cylinder piston engine. See You Tube 14.5 hp B&S hard to start/ valve adjustment.
Leonard Bonamo, August 2013