My own approach in an emergency situation where loss of aircraft was a possibility, I would take control and use crew resource management techniques to include the student as a resource to share tasks with. If loss of aircraft wasn't a concern I'd let the student troubleshoot and make appropriate decisions but as soon as that line was crossed, I'd assume control (per the preflight briefing), and would handle the situation as the actual pilot in command using all available resources including the student to safely recover the aircraft.
Genghis, It sounds like you did the right thing including the thorough post-flight debrief. A brutally honest postflight debrief after an inflight emergency can make a huge and lasting impact on a young pilot's career. If you make a mistake, own it and debrief to it. I've had students call me out on areas of discrepancy that I glossed over, and every time I let them know that I approved of their thorough and honest debrief inputs and spent extra time going over whatever questions they may have had. I think it helped, they're all still alive anyhow.