Contrary to your assertion the two engined craft are just as safe (or safer!) as the four engined craft because of changes in procedures. Fuel management, weather forecasting, general engine reliability, lubricant reserves are all different and much more stringent for a twin engined jet over water on a straightline route. The effect is the aircraft is less long in the air anyway which has a big effect on the risk calculation.
The main motivation is money, that's true, but the secondary motivation, the environment is also important and the math really does work out to a negligible safety difference (advantage twins) that to date has not led to the observation of numbers that would indicate a deviation from the predicted reliability figures.
Finally, the chances of something going wrong on a twin engined aircraft vs a quad are higher for the quad because it is a much more complex system. So yes, if an engine fails a four engined plane would be able to deal with it better but the chances of something going wrong are twice as high.
In the 60's, when the 747 was designed the reliability of jet engines was such that four engines was more or less a must but we are now a good 50 years away from those days and our ability to deal with complex reliability issues has gone up substantially (recent Boeing issues notwithstanding). The A380 has four engines because it is simply too heavy to take off with only two engines.
Engine shutdown is so rare now that pilots can go a whole career without encountering it even once and typically it is the monitoring system that has a fault rather than the engine itself but you would shut down the engine anyway in those situations as a precaution.