if Chad does not cooperate, it becomes difficult for Sage to convince everyone that everything is fine between them
if nobody ever sees them together, never in intimacy, gossip would still start to circulate (regardless of MC)
Sage is a classic co-dependent person. We see it acted out glaringly in the flashback when she is chasing after Chad trying to be optimistic about the Halloween party and he's clearly disinterested, so this unhealthy relationship has been going on for years (since before the formation of the DIK). She needs to dump him but that side of herself won't let her. That's why she wants to "find the woman he's sleeping with." She can somehow blame "her" and keep "him", even though deep down she knows that's not how it works, because the idea of dumping him isn't a big part of her self-talk. CD runs deep and is often heartbreaking, especially when the other person is abusive or, as in the case of Chad, just disassociated. Of course, he's a clown because he treats her like used socks and gets outraged if he even thinks someone is looking at her, even though he can't be bothered to give her the attention she craves.
Poor Sage. In many ways though, the MC is helping her to see that she is valued and how she should be treated. We'll see where the story arc takes her in the next Episode.
Once again, DrPink is masterful at characterization and motivation. Great writing.