Is the buffer in this power supply circuit required?
There is no safety concern, nor "impedance matching" (since we don't know any load characteristics). If you use just a 100k trimpot, the current carrying capability of this "power supply" will be about 20 uA (micro Amperes). The LM358 buffer allows this power supply to supply 20-30mA of current at stable -1.25V voltage level. Not much per power supply standards, but still quite more than 20uA.

ADDITION: If it is required only 0.2mA and cost is of concern, the most economical solution could be to use "parametric regulator" using a 1.7-1.8V Zener diode and 30k resistor, with a ~5k trimpot to adjust the voltage to -1.25V, assuming 15-20% of voltage tolerance and not much of load variability.