Function of the FET in LM350/LM317 programmable current limiter
Yes that is a FET but what kind of FET ?
It's a JFET and these have the property that some current flows when Vgs = 0

Look in the 2n5640 datasheet, you will see that IDSS, the Zero-gate-voltage Drain current is around 5 mA typical.

So this JFET simply behaves as a current source of roughly 5 mA.
Ideal to power a LED at constant current :-)
Or make a cheap-ass 1.25 V voltage dropper with 2 diodes like in this schematic.
As long as Vout - Vss is large enough for the JFET to behave as a current source, this will work.

The JFET could be replaced by a resistor but then the circuit would become more sensitive to Vout - Vss.