Nothing to get so worked up about. They probably incorporate in Singapore for tax purposes. Neither SIA nor any Singapore government-related entity or even any Singaporean owned or controlled company has a significant stake (if any) in Skywest.
Rex, however, is owned by a group of Singaporean investors (private ones, nothing to do with SIA, or the govt).
If ultimate control in Skywest was really in SIA, as a listed company SIA would need to disclose this to the market and consolidate Skywest's accounts line by line into SIA's Group accounts.