The breaker is NOT the only, or even the most important consideration regarding inrush current.
The inrush for a given voltage is dependent on the resistance vs voltage. For a switching power supply, there is no particular limit aside from the ESR of the capacitors. You will NOT get the lower inrush by setting to 120V if you have it operating at 240. You WILL get other problems, however.
Putting a regular NTC inrush protector in series, if it is properly rated, will definitely help with the inrush current. It MAY cause operational problems, however. Maybe less so with a switching supply than with a transformer based supply.
The problem is that the inrush protector has a relatively high resistance when cold, to limit the current, and goes down in resistance when it heats up. A properly rated one will go to a fairly low resistance with the "normal current" in the circuit.
But, if the circuit has intermittent high currents, as a stepper supply may have, then the protector (the "NTC") may be allowed to cool off and go to a high resistance. At that point, a high current drain may lead to a drop in voltage, which can be enough to bother the circuit. If you have a switching supply, that is rated for any voltage from 90 to 264, as some are, that may be no issue at 240V (might be a problem at 120V).
But if the supply is switchable, either to 120 OR 240, then it has little reserve, and the voltage drop may be an issue, the supply may not be able to maintain output voltage without sags. With transformer based power supplies, the voltage drop is even more of a problem, and can lead to problems in operation of the steppers.
Inrush protectors also have ratings which must be taken into account. They are often rated for the connected capacitance and voltage. The capacitance tells the energy (current * time) that the thing has to handle at the operating voltage. If you have a transformer in series, that can be difficult to figure, but with a switching supply you know tha capacitance. If you do not use a properly rated part, it will fail, probably failing "open".
As for the real problem.... The rectifiers and capacitors in the power supply take a big hit when the inrush occurs. That shortens life, and can lead to a failure. The capacitors "do not like" huge current surges. Rectifiers have limits also, although they often can stand more.
So there are reasons for using some form of inrush limiter even if the breaker is OK. And there are reasons for using a relay to switch the protector out when the power supply voltage has come up.