Yagudin looked down on Plush for many reasons. Plush was a "provincial" adolescent - all peterburgian students looked down on him. I also think Yagudin is more willful and had more of a problem with Mishin's overbearing style; at the same time, Mishin had more of an issue with Yagudin's willfulness. Evgeny saw a father figure in Mishin; Alexei did not.
Mishin/ TT rivalry goes back a long way. TT was much more of an establishment person than Mishin. Due to her father's fame, she was able to land Rodnina and Zaitsev for her students when she was just starting out as a coach. Mishin, OTOH, for many years couldn't even travel abroad with his students (there was some intrigue, someone claimed he was a flight risk; he now says he know who it was who made trouble for him). Furthermore, Mishin always had some disdain for ice dancing coaches and the way they (Tchaikovskaya, TT, Dubova) always bickered and back-stabbed. Finally, when TT started to coach singles, Mishin didn't appreciate it, and saw it as something artsy-dancy replacing the discipline with jumps etc.
Also, I believe the media really overplayed the whole thing. BTW, I don't think TT has much of a problem with Mishin.