Remember that fixed Voltage active devices like batteries and Diodes and power sources must all have an Effective Series Resistance (ESR). We can't always ignore it and use Ohm's Law to determine the current flow when including these devices are in parallel and must add external series R's to limit the current.
To compare passive (diode) vs linear active ( feedback loop) performance for regulators, there are 2 basic attributes. Source/ Load regulation error and ESR
In all regulator methods, the" load regulation error" on source voltage is important as the range of output voltage over range of output current. THis is also the ESR of the source.
The % of load regulation error is defined as the ratio of Source ESR/Load Impedance.
If the load is static, then the error is DC only, if the load is dynamic, then the impedance ratio induces a % of output DC as AC drop or ripple we call Load Regulation Error.
All Passive diode regulators must draw more current than the load to remain saturated. This can cause all power consumed by load to be dissipated in Diode when there is no load and thus a potential thermal and efficiency issue.
In Active Bandgap Regulators the idle current is only the minimum required to saturate the diode and bias the buffer, which for FETs can be very low current and much lower dropout than bipolar Darlington types with 2V drop.
The ESR of the Zener, LED or silicon Diode can be predicted as the inverse of the Max continuous power rating of the device. ESR= 1/Watts is a good approximation for If>=/10% of max current.
Bandgap Reference Diodes are very temperature stable by design but must be buffered since they can only supply <=10 mA
To use a White LED's as a 3 Voltage Regulator, assuming your bias current is < max rating and greater than MCU load. Then LED will dim when it consumes more current at max clock rate or after waking up. Depending on size, this can range from 20mA 5mm types to 1A 3Watt types.