Belt and pulley fell off 40" attic fan & my repair failed (updated)
Check the capacitor

Since the motor starts and runs no-load, it's not the motor windings themselves, and the capacitor isn't open or shorted most likely.  I'd check it anyway though, as it's likely way out of spec, which'd explain the behavior you're seeing (starts unloaded, but fails to start up under load) and the motor is a Permanent Split Capacitor (PSC) type, so there's no start switch involved here.