Is there an advantage of reducing the single sideband carrier in amplitude modulation over supressing the carrier completely?
Does it have anything to do with demodulation?

Leaving a small bit of carrier energy is entirely to make demodulation a lot easier. If you can extract the carrier from the transmission you can synchronously demodulate the single sideband back to baseband relatively easily. It's also very similar to vestigial sideband modulation and here's a reference for Full, reduced, and suppressed-carrier SSB.