But the turbine generates power to keep the plane flying. Why would it only work for 10 minutes? Certainly the flight time is a product of fuel level and altitude. Even if both engines fail the flight time would be a function of altitude. I don’t see how deployment of the RAT informs flight time.

It does not generate power to provide thrust; it generates power - using the airstream as the aircraft moves through the air - for the avionics and/or hydraulics.

Yes, exactly.