I think Chad does care about Sage to an extent, just not romantically. He was in the relationship for appearance (cover for being gay is the most likely explanation, but there are other possibilities too), but had no way to leave it. Mel obviously knows the story and suggested this would be the best time to end things. That doesn't quite explain why Mel (or Chad) would pick now if the MC isn't pursuing Sage, but I think it covers most of the rest.
I do take your point, but we've seen Chad quite a bit and he's not usually so... tentative when interacting with people. To me that makes it seem like he's not being honest during the breakup. If you think otherwise, we'll have to agree to disagree.