Honestly I would expect Yang to give Sullivan a thrashing if he actually tries this crap. China is not America's messenger.
Sure, Xi could brief Putin on what went down in the Yang-Sullivan talk, but if the US has a message to pass along, then pass it to the Russians, directly.
It's not China's problem the US finds it shy/embarrassed/awkward to talk to Putin, nor is it China's responsibility to "clear the air" cause two dudes ain't talking to each other, especially at the same time one of these dudes keeps flipping your lunch trays and blowing spitballs at your face like a right a**hole, and now he expects you to mediate?!
Not to mention the blatant disrespect of diplomatic protocol. Where does Sullivan, a National Security Advisor i.e. glorified aide to the President's office, get off thinking he can deputise China's highest diplomatic officer to tell the leader of China what to say to the leader of Russia?!
The level of arrogance, I'm tellin' you.