Buit if the links I found are accurate about Saturn’s core temperature, and since Saturn is assumed to be mostly gaseous, the core temperature , if around 7000°C would igniote the gaseous atmosphere and melt all liquid or ice into gas, ignote them...
Saturn should be on fire if this is true about the core temperature.
Same with jupiter. AND Uranus and Neptune
The core temperature wouldn’t affect the atmosphere. Our own core temperature is 6,000 degrees, and Saturn’s diameter is about 19 times that of the Earth.