Benchtop power supply current limit on MC34063 switched-mode power supply
I can not really figure out why the regulator does not limit the excessive current if you start with high current values. Maybe the current does not actually flow through the current sensor?

But I have a suggestion for a simple soft start.







simulate this circuit &ndash; Schematic created using CircuitLab

Consider the capacitor C5 between VCC and Input. On power up this will pull the input pin to VCC wich is higher then 1.25V. So the comperator inside MC34063 will not turn on the switch.

C5 will now be charged by R1 with a time constant of

$$\tau = R1 \cdot C5$$

For a current of 1A, R1 should aproxamately be

$$R1 = \frac{1.25V}{1A * 51 m\Omega \frac{124 µA}{30 mV}} = 5929\Omega \approx 6k\Omega$$

$$\tau = R1 \cdot C5 = 6k\Omega \cdot 1µF = 0.006 s$$

And after \$5\tau = 0.03 s\$ the capacitor will be almost fully (99.3%) charged.
During this time the feedback of the current sensor will slowly take over. Once the Input voltage goes under 1.25V the PWM will start.

The only thing that is not quite nice with this solution that it might violate the maximum conditions of the ZXCT1109. It states that the Voltage between OUT and GND may not exceed \$-0.3\$ to \$V_{S+}\$. Since the diode D1 shout take care that the Voltage at S+ is almost the same as VCC this might still work. 

To not violate the specifications do not use the capacitor as explained in position C5 but rather connect it as C4 between OUT and S+. This will work almost the same.