My two cents.....
Most diffused RED LED's are GaAsP on GaP is 1.4V +/- 5%** + ESR*Id, which could be as high as 2V, but at low current such as the original cct. with < 0.5mA it will be at the minimum voltage. 1.43V measured is <3% of nominal tolerance.
The same for silicon transistors at low current it will be Vbe= 0.6V +/-5%** +ESR*Ic which can rise to 0.7 at moderate current and 1V at high base currents. Since Base current is Ic/hFE and collector current is low, base current will be minimal drop or 0.6V.
Verification
Thus the emitter current = 1.4-0.6=>0.8V/39Ω= 20mA ~ +/-15% incl R tolerance of 10% = 17~23mA. measured 16.93mA is at the low end of estimate.
** at room temp.
The modified transistor circuit will cutoff at two Vbe voltages for the main driver or 1.2V for low current and have a slope of ∂V+ / ∂Ic = Rb/hFE on the constant current of Ve/Re=Ic, which only becomes important ,if say the supply voltage increases. So it is pretty constant but increases by base current / beta.
Low Vbe is not a problem but supply regulation of constant current but it is not perfect as it is on the cusp of the Vbe curve but not much better than the LED for regulation.
To scale this design to higher current LED's such as 300mA with different transistors, a similar LED used on the base will be a more effective regulator as 1W LED's have an ESR of ~1Ω, and a 5mm LED@ 20mA has an ESR of 10~16Ω, which is much less than an zener or a transistor Vbe. So LED's make great Zener type regulators when driver past 10% or rated current. Here 0.5mA is a little shy of 10% rated current, so regulation could be improved but was adequate.
If you wanted to have tighter control of LED current, one would need to use a precision band gap reference such as in LM317 1.25V across ADJ and Output and put current limited resistor across those two terminals. THis is even simpler than the above circuits but all suffer from some effective voltage drop across the current sink/source of 2V min at minimum supply voltage.
Output stage
As this is a source follower is like an emitter follower, the source DC current must exceed the AC peak current in the output AC coupled load resistance of 500Ω at maximum swing to remain linear.
If you need to increase voltage swing, simply increase current sink value using parallel R's and keep in mind this method is inefficient but necessary for linearity, so Q may get hot and need attention for choice of device.
I hope this makes sense.