That's a peculiar way to flatter someone, and even then, Sensei just ignored it. He wasn't flattered. Ayane also ends up being surprised by Sensei sleeping with others, not noticing Molly was jealous of Otoha, not doing anything when Maya cried, and she doesn't really seem to understand anyone when it comes down to it. Ami, Makoto, Maya, Sana, Sensei, etc. Keep in mind that she thought Sana was completely innocent and pure, when that's just an act. Her answer to Ami's problems is also simply violence.
Empathy requires accuracy or it's not actually empathy. It's about understanding and feeling what others feel. Overcoming it together. Being wrong about what someone is feeling, shows a lack of understanding. Ayane thinks she understands others, yet never really does, which speaks for itself.
That card was actually Chika's idea to get Sensei to come back. Ayane mentioning others, especially Maya, was just to get what she wanted in the end, Sensei back. She's definitely not trying to get Maya and Sensei together. Ayane was simply out of ideas and was hoping for a miracle the night before.
Ayane also can't stop others from pursuing Sensei. Sensei goes to them, and even if she tried, that'd mess up her relationship with Sensei. That hasn't stopped her from secretly slapping Ami, trying to cuck Maya on the roof, and fucking him on both Ami's and Sana's beds behind their back, though.
Sympathy and empathy are different things. For example, Makoto isn't really bothered by Maya, like Ayane was. Ayane can understand being belittled by Maya, but not how being belittled by Maya actually makes others feel.
Ayane's kind of earned Ami's disdain (most have at this point). Lying to her, intentionally leaving her out of things, trying to fuck the man she loves on her bed, etc.
Early Chapter 4 Ami reached her breaking point, which has likely been building up since Sensei let slip he carried stuff at night for Maya back in Chapter 1. At least. From Ami's point of view, her friends have continuously been betraying her and endangering the only family she has left, for a long time, behind her back, and nothing she does can stop it or him, from hurting himself and her. It's not just Sensei who broke her. Everyone's been hurting her:
Imagine if your best friends were sneaking around, fucking, and endangering your love interest that you do virtually everything for. One wrong move could lead to them being locked away. That's more or less Ami's life.
Just add voices, delusions, blackouts, more or less raising that adult love interest as a kid, and ptsd from watching your parents be taken from you, on top of it. This world also likely plays a factor. She seems aware of how things don't make sense, and are usually repetitive.
Overall: Ami has been driven insane and recently reached a breaking point. Ayane helped her get here.