Bandgaps are noisy. Recall the LM723, which had a buried zener and was recognized as one of the lowest noise regulators available.
I spent considerable time looking for a zener model that included noise without success. However, a zener has to be quieter than a bandgap.
Here is a circuit that uses a 1N751 in a variable output voltage regulator.
It has decent frequency response, -28dB at 1mHz, with a slight peak at 6 Hz,
-80dB at 120 Hz, -165dB from 100 KHz to 3 MHz, then falls from there. It probably would be much quieter than the Motorola TL431, and uses standard junk box quality parts.