I'm not 100% convinced, but for the sake of having all the facts I found two things that could support your theory:
Firstly, after our tennis date with Jill, she posts on Rooster about it. Tybalt leaves several cringe posts, as usual. Then Sage, out of nowhere, responds by saying "Give it a rest, {Tybalt's tag}". He tags her back and immediately says "This is hardly any of your business!" This is significant because, unless I totally blanked on something, this is the first time Sage and Tybalt have interacted at all. Not only that, but I don't think Sage and Jill have interacted either, so why would Sage chime in on one of her posts? Maybe instead she was following Tybalt, if he is her brother, and got tired of him shamelessly sniffing after this girl? I can easily read that exchange as one between siblings.
The second thing is the wording red-headed guy uses when talking about his daughter: "And can you believe that I've yet to find my daughter in this crowd? I was under the impression she'd attend the party." Why would Sage's father expect to see her at a prep party? She clearly doesn't associate with them, and nobody knew the HOTs would be there. On the other hand, if we buy your idea above, red-headed guy could be Jill's father. This would explain why the businessmen are polite to his face despite clearly disliking him, and why the prep students want to impress him: he's a Royce.
Again, I'm not sure I'm 100% behind the idea that Sage is secretly a Burke, but I can't dismiss it out of hand...