I cannot think of an easy way to do that. It is difficult because when the power is switched on, the Arduino does not know what the current position of the servos is. That current position would be needed to make a slow movement towards the start position. Without that, the servos will move at their normal, fast speed to reach the start position.
If the Arduino had sensors to inform it of the position of each servo, it would be possible. Maybe replacement servos are available which will allow the Arduino to read each servo's position, I do not know.
Another possibility is to use the Arduino's eeprom memory to record the servo positions as they move. Data in the eeprom memory is not lost when power is switched off. When power is switched on, the Arduino could read the eeprom to find the servo positions and move them slowly to the start position. One problem with this idea is that the eeprom' life is used up a little each time data is written to it, so it will not last forever.
An on/off switch could be connected to the Arduino. This switch would not cut the power. The Arduino would read the switch, and if it has been changed to off, move the servos to their start position before going into a low power sleep mode until the switch is changed to the on position.
None of these ideas are simple or easy. Maybe they are not worth it to fix the startup problem.