Yeah, I rarely pursue Jade and almost never tell the DIKs if I do, but it's a good point. Even if the DIKs are supposed to forget about it, you'd think mentioning that the hot teacher we fucked is also Sage's mom would be a top priority here...
The MC's aversion to last names is frustrating, but it may be justified in Sage's case. We know she's uncomfortable with her heritage, and she joined the HOTs in part to get away from it. It would make sense that she'd actively hide her last name (or use another one entirely) to hide her connection to the highly visible Burkes. The MC doesn't share any classes with her, so he wouldn't see her full name on official documents or the like; it wouldn't take much effort for Sage to use a pseudonym with him.
Still doesn't excuse Jacob not giving the MC a heads up, or Sage herself if she's bringing the MC to meet a family she knows he has bad blood with.