“Why do you think the minhag became not to?”
I dont know.
The Agur says because they would go to the Mikva before Duchaning but was too cold.
The Remah says becasue lack of Simcha.
Others (I forgot offhand who) say becuase Kohanim arent muchzuk bizman hazeh
The real reason may very well be something else.
The bottom line is as the aruch Hashulchan says (and I am loosely translating (128:64) “Certainly there is no good reason to our minhag to be mevatel a mitzvas aseh of Birchos Kohanim during the year, they write it is a bad minhag “Minhag garua hu”… ”
“How did the minhag develop based on the justification? Until the change, there was no minhag to “trump” the halachah!”
thats a great kasha, (it is similar to your above kasha) Thr truth is I dont know.
I would venture to say that as the minhag started it was keneged halacha and they had no right to be mevatel a mitzvas aseh. With time, it became accepted for whatever reason and now we rely on minhag avoseinu with justification from Gedolei Acharonim to “trump” the mitzva of Birchos Kohanim. (and it seems this is possibly ratzon shomayim as seen in the Stories when the Gra and R’Chaim volozin tried to reinstitute it)