The LM317 requires a minimum difference of 3V between input and output. It can't provide any kind of regulated ouput when the input is only 3V.
If the input were higher, you would need a negative voltage source to get down to 1.25 V. The LM317 uses a reference voltage of 1.25 V. The ouput voltage must be above 1.25V, else the regulator can't operate. The datasheet gives a circuit using a negative voltage supply to change the operating conditions so that you can regulate the output right down to zero volts.
Probably simpler and cheaper to get a modern part that is intended to work for your case than to try to make the ancient and venerable (and decrepit) LM317 do the job.
All of the things I've pointed out are mentioned in the datasheet of the LM317.
That's the reason you will see the often repeated mantra: "Read the datasheet."
The softstart works by pulling the adjust pin to ground through the PNP transistor.
A grounded adjust pin forces the LM317 to output its minimum voltage.
The base of the transistor is held low by the (discharged) 10uF capacitor.
When a little voltage appears at the output of the 317, the capacitor starts to charge. The voltage rises, and the transistor slowly starts to shut off.
This allows the voltage on the adjust pin to rise - the output starts to head towards the set output voltage.
At some point, the transistor shuts off completely, and the output is at the set voltage.
The length of the soft start is determined by the 10uF capacitor and the 50k resistor.
A larger resistor and/or capacitor will lengthen the softstart.
Smaller values will shorten the softstart.
It is actually the product of the resistance and capacitance that control the time, so you could increase one and decrease the other and come out with the same time.
You also aren't completely free to choose just any values you like. There are limits to what values you can use, but I'm don't know how to quantify it (and don't have much interest in figuring it out.)
The soft start would probably work if you had a supply voltage high enough (at least 4.3V) and you were using the negative voltage as described in the datasheet.