An engine system includes an internal combustion engine, a main exhaust aftertreatment system with a main catalytic converter, and a light-off catalyst bypass system with a bypass passage and a bypass catalytic converter. An emissions control system includes a controller in signal communication with an oxygen sensor disposed downstream of the bypass catalytic converter. The emissions control system performs a diagnostic of the bypass catalytic converter, by the controller, including (i) monitoring signals from the downstream oxygen sensor for a predetermined time period during an engine cold start condition, (ii) determining a curve plotting oxygen content at the downstream oxygen sensor over the predetermined time period, based on the signals from the downstream oxygen sensor, (iii) determining a number of times the curve crosses a predetermined setpoint over the predetermined time period, and (iv) comparing the number to a predetermined threshold to determine if the bypass catalytic converter has failed.