In addition to Daselbee's above first paragraph on target statement (Post 560), the stator under the flywheel is a two fold component...... there are a series of small coils that pertain to the charging system, and two larger black coils that pertain to the ignition system.
Those two large black coils provide approximately 300 AC volts to the capacitor within the powerpack, needed to energize the powerpack and provide ignition.
The series of smaller coils may be perfectly okay, providing charging power (obvious as it powers your tachomter) BUT if the regulator/rectifier is faulty, that charging power cannot pass thru to the battery.
As far as the engine running with one cable removed (excellent way to damage the charging system)........ Even if no battery was present, and with the key in the ON position, and if one was in a physical condition whereas he/she could crank the engine over fast enough to have the stator energize the powerpack, the engine would start and run. Think about it..... your engine has the same type ignition that the smaller (non battery) models has.... and they start simply by cranking them over via manual start.
All of the above posts/replies spell out Faulty Voltage Regulator/Rectfier. The mechanic that stated "if I remove ground cable from motor it will stay running and that proves charging system is working" needs a refresher course in electronics.