Coax, S-Video, and Composite inputs are old input types and the TV’s CC circuitry functions as normal.
If you have a High Definition TV showing a picture using the component input (Y/Cr/Cb), the TV’s CC circuitry is disabled. This is apparently due to the digital CC specs giving the source of the signal (whether it be a HD receiver, DVD player, or DVD recorder) the responsibility to display the captions. My HD receiver has CC circuitry built in and it works. My progressive scan DVD player does not.