If you have full flaps in a C172 (Older 40 deg, newer 30 deg) and are on speed and still wish to steepen the approach, it must be an "interesting" approach or you have miscalculated.
My understanding is that Ratherbe flying is correct. The flap airflow disturbes the elevator airflow, leading to "oscillation", which disturbs the pilot, which disturbs Cessna's lawyers. Be aware also that some manuals caution against sideslips on low fuel because you can uncover the inlets.
POH's for American aircraft seem to be full of legally approved lame generalisations, like Piper's "it is good practice to contact the runway at the slowest speed possible."
The switch from 40 degrees to thirty degrees was made to increase payload - its related to the aircrafts ability to maintain height at gross weight. I found this out when a flap switch stuck at 40 on a go around in a C150. We never exceeded fifty feet and only just made it back to the end of the strip.