There was a very good and well-researched article by Tony Hoskins, "Mary Boleyn’s Carey Children: Offspring of King Henry VIII?” _The Genealogists’
Magazine_ (London), Vol. 25 (March 1997) that was well received by historians and genealogists alike.
From memory, it was based on dates, and extensive grants to William Carey (I think) that had to pass to his children with Mary Boleyn. Queen Elizabeth I also I think gave them lands and position at court and built both Henry and his sister Katherine magnificent tombs in Westminster Abbey. I think this was seen as Elizabeth knowing that they were her half-sister and brother, but the fact that they were first cousins and cousins who had no claim on her throne, may have been reason enough.
Although there have been a few other articles since Tony Hoskins, that Henry VIII was their father will never be proved conclusively (unless there is eventually some DNA evidence), and I'm inclined to retain William Carey as their father on Wikitree, with a note in their biographies pointing out Henry VIII may have been their father.
I think the point that Henry Carey is named Henry, isn't a strong argument; Henry was a common name in England and with a powerful king like Henry VIII you are probably going to want to flatter him anyway you can.