Tumble Dryer Capacitor
That device is a line EMI (R.F.I.) filter P/N 4213 078 52551, not directly related to the motor operation, despite the similar packaging to some run or start capacitors. 

It's a separate part from the motor run capacitor and should be checked too (it could go open). But it sounds from the symptom (runs when hand-started) that it's the 8uF run capacitor that has lost capacitance or gone completely open.