How did the black frock coat come to be garb of some roshei yeshivos and other senior rabbis?
It all started way back in the days of the Maharik. See Shoresh 88 which is pointed to and loosely quoted in the Ramma Yoreh Deah 178 siff two. This same responsa is brought in hilchos Tzitzis siman 10, as well.
Apparently professionals of any certain field began wearing a special long coat called a Kappah (sounds similar to kappata, no?) and the Rabbis followed suit. The Maharik was questioned if this is allowed or perhaps this is an issue of going in the path of the idolatrous. The Maharik went into a long explenation that following the path of the idol worshippers is only a problem when the behavior being mimicked is less 'tznius' than the practice of the Jews. His use of the word tznius means humility as is understood there, as well as the common usage of the word nowadays. He therefore allowed this Kappa as it is not a haughty showing but rather a showing of professional status, as opposed to wearing red which is a color reserved for big shots and therefore lacking in the tznius a Jew is supposed to have. It is also not less tznius being that it is longer, rather than shorter than what people wear.
Ironically he lambasted the questioner for even questioning something that all the great Rabbis were doing.
I would definitely suggest reading the responsa inside. I found it enlightening in light of what people think the Ramma there is saying.