“Based on this discussion among the CR poskim, why is that the suggested method?”
CR poskim? You mean like the Rema, Mishna Berura, Yechava daas?
Yes, it is better to avoid lechatchila using the same keilim. Tht ist the case you mentioned. Note: you accuratly quote the shemrias shabbos Kehilchaso as saying “one should TRY” ifnot practical there isn no problem whith using the same kli.
The part you left out is the footnote (181) where he writess that after 24 hours the kelim are mutar (without kashering)