Why Did Mordechai Care About Bigtan and Teresh’s Assassination Plot?

To prevent chaos (not anarchy!), see Bava Metzia 83b.

Also, the Medrash says they were planning on pinning the crime on Esther.

See also Kedushas Tzion issue 41 page 6 by Rabbi Goldberg, that Mordechai’s plan throughout his stay in the diaspora was to convince the king to rebuild the temple.
(Forget the wisecrack by Rabbi Yehonasan Eibeshutz…)