His serial cheating is a big part. I don't think it's just about getting involved with students, but the fact that he or the student is often married.

He was supposedly married, and his wife pregnant, when he got involved with Shae-Lynn. Then while married to Shae-Lynn, had an affair with Grushina, while coaching her
and her husband. Kind of awkward.

With that said,
I don't like how others like Zhulin seem to get more of a pass for doing similar things (cheating on Usova with Navka and Grishuk, and I'm pretty sure Mikhailova was still in juniors when he started dating her and I believe he was still married to Navka too). I think Morozov's generally creepy personality and his racist comments make people less inclined to like him.
My main skating issue with him is as an ice dance coach/choreographer. With the singles and pairs, yeah, he mostly sucks, but not always, and he is hardly the only one choreographing empty programs. I have higher standards for dance. So many of his dances were carbon copies of previous ones. GrushGons 2005 FD ripped off L&T's Air/Storm and their 2006 FD ripped of G&P and N&K's The Feeling Begins. GregPets 2006 FD was a ripoff of A&P's Romeo & Juliet. I'm not saying you can never use the same music as another team, but those dances included similar music cuts, choreography, and costumes. It was just lazy.
And it's not much better when he tries to be original. B&K's 2003 FD was like a parody of Russian ice dance, but not meant to be. Then there was I&K's Ave Maria where they waited to reveal it (going so far as to skate it to different music at the Russian test skate) because it was so original... and, well, let's just say there was a reason noone has used it before.

And then this year, outdoing himself with Ghost... I mean, they literally have the words "I'm talking to YOU, esse!" in their FD! I just can't!