You also have to realize that in stick and rudder they are likely talking about J3 cubs or luscombes or old T Crafts, not your run of the mill trainer as recent as the 70s.

In todays modern planes if you find yourself high on final and are on speed you can simply "dump" a little more flap to increase your descent rate while maintaining airspeed with pitch or power (your choice).

"Mushing" (what a stupid word) is truly a back country/short field technique. You can come in higher than normal and get "closer" to the field bypassing the 3 degree glide path and come in closer to a 6+ degree glide path also known simply as being high on final. Once you hit the designated point where you are high you can cut the power back a bit and raise the nose to a near stall airspeed and then maintain your glidepath with power riding the backside of the curve the whole way down. Finish off with either what little flare you have left just before touchdown and/or a small burst of power to arrest your descent. This nearly eliminates float as well as puts you on the ground with little forward energy at the slowest speed possible.

The same technique is what they are mentioning but they mention it in the book merely as a way to get down when you are high.

Personally in a J3 or flapless airplane if I have no need to use less than 400 feet of runway I will use an appropriate well timed slip which will do everything I need and then some. In a slip it is very easy to control both descent rate as well as airspeed and is a bit of an ideal technique IMHO.

As a primary student, first off all, kudos for reading that book. There are some great things in that book that still ring very true today. Take some of it with a grain of salt though as they may not be an ideal technique in todays modern realm of flight. I would not recommend any student raise the nose and mush down final anywhere near stall. I would push my student to simply fly a configured approach on profile and instead of teaching them to save it I would teach them to go around. You can learn to save it as you gain experience.