What is a variable speed drive (VSD)?

In theory variable speed drive (VSD) is able to reduce the motor starting current at very low values is a fact that motor has limitations. I have seen many motors to stall because someone limited the starting current to a very low value. That low value is an argument to sell the equipment. Limiting the starting current depends on the type of motor and load operation curve reflected at motor shaft, for example a pumping system has a different curve than a conveyor and each application needs different starting torque and current.

At startup, a motor's backEMF is (near) zero, so a Direct-On-Line start causes a large current to flow (up to 6x rated). As current translates into torque, this also gives high starting torque.

However, in cases where such a high starting current is undesirable (such as high powered motors), the current can be reduced in two basic ways. The first is to increase the impedance, achieved using resistors in series with the motor during startup, switching them out during normal operation to maintain efficiency. The second is to reduce the applied voltage to the motor, i.e. star/delta startup.

An alternate approach is to actively regulate the current into the motor, which is what a motor drive does. It uses controlled switches (thyristors, IGBTs, etc.) to manage the flow of current into the motor, which gives it the ability to limit startup current. However, as torque is still directly related to current, the starting torque must also be affected.

All variable speed drives have soft starting as an inherent feature & thus they will definitely reduce the starting kick of motors.

And, variable speed drives will reduce current requirement of motors provided the load is less. But, that could happen without drives even. What variable frequency drive do is energy conservation, if the process flow requirements are variable.

But while you can use a variable speed drive to soft start a motor, it's expensive compared to a solid state starter if you intend to run the load a full speed all the time and there is no need for flow or speed control. A solid state starter achieves the same current limit and reduced water hammer effects by controlling just the voltage to the motor, typically using thyristors. Many soft starters have a contactor that bypasses the thyristor bridge once the motor gets to full speed.

One essential difference between a soft starter and a variable frequency drive in this regard is, that the variable speed drive delivers "nearly" sinusoidal voltages (and currents) to the motor, which makes it possible to develop high starting torques during the acceleration, even higher than nominal full load torque, depending on the application, while a soft starter only supplies fractions of the basic waveform, which serves to reduce the current to the motor significantly, but still at the nominal frequency. This will reduce the available starting torque dramatically until the motor is up to around two-thirds of nominal speed, or maybe even higher.
There is very advanced version of this which is called a cycloconverter, but that is even more expensive than a variable speed drive.