Right - so we're all in agreement, then!
BTW, at our little school (flying the mighty Schweizer), we demo EOLs to students before they solo They must be able to perform "approaches with the engine idling", followed by "flare recoveries", as well as power failure at the hover, aka "hover autos". They must also show proficiency at dealing with simulated engine failures to a go-around. Once post-solo, if a CFI feels a Private student will benefit from EOL practice, they are authorized to provide that training, after an O.K. from the chief (that's me). At the Commercial level, all students recieve EOL training, as well as training in engine failures on approach, departure, OGE hover and so on.