OK, I'll put that down to "phrasing"...
If a girl, who really likes you, comes to your house and tells you she's organised a lovely day out for your birthday, and she (or one of her many servants) has made a picnic and everything. Do you think she'd prefer you to let her down to go see to a sick friend who is another girl (yeah, the mc lets on that it's a girl he's seeing), or that you go out on the picnic with her (and that she doesn't even know about the sick friend)?
Logically speaking, the mc already made his plans with Sage, so he hasn't done anything wrong by telling Jill:
mc: "I'm so sorry. You caught me at a bad time."
mc: "I made a promise to a friend to come help her."
mc: "She's sick and needs someone to take care of her."
Now I know this game has DIK/Chick points, and that'd be the chick thing to do, and Jill likes Chicky boys, but tell me honestly, which response would a real life girl prefer? Of course she would prefer that you go with her.
However, if you told her that you'd already made plans, but you're happy to ditch the sick girl, that would complicate things.
A girl would want you to think they are reasonable and would say, "Of course, you should see your sick friend", but she'd want you to say, "You know what, she'll probably be dead by the morning, so there's no real point seeing her now anyway" to which she'd reply, "Are you sure? ... Yaaah, let's go!"