By limiting current on turn-on, soft start circuits greatly extend the life of tubes. When cold, the filament in the tube has a much lower resistance than it does at operating temperature. That means that when you turn the component on, there is a sudden surge of current and very rapid heating of the filament. It is this thermal shock that is bad for the tube. If the amp has a tube rectifier, this acts as a form of soft start for the rest of the tubes.
Therefore, a soft start circuit (e.g., a series-connected thermistor) is most useful in an amp with solid-state rectification. The thermistor will not fully conduct current when it is warmed up, so it can, to some extent, reduce the current flowing into the amp after it has done its job. I personally don't know how much of a sonic effect that will have, but this is the trade-off.
Therefore, a soft start circuit (e.g., a series-connected thermistor) is most useful in an amp with solid-state rectification. The thermistor will not fully conduct current when it is warmed up, so it can, to some extent, reduce the current flowing into the amp after it has done its job. I personally don't know how much of a sonic effect that will have, but this is the trade-off.