OZ doesn't know or care how many miles you paid for your ticket. They know and care what UA is paying them, and they're getting paid for HKG-ICN in C and ICN-ORD in F, since that's how the ticket was booked.

You would have to get UA to move you into F on HKG-ICN for OZ to care.

And as others have pointed out, once travel begins on a Star Alliance award ticket issued by UA, no changes are allowed. If this was the first flight on your ticket, UA should have been able to make the change, provided OZ had O inventory available on your HKG-ICN flight.

It's unfortunate that you were given bad advice by the phone reps, but the OZ agent was right.