Ah, so I am not the only one who tried this

Wasn't sure if this was in any way a good idea or I just had a strike of luck...
Anyways, in May 2021 I did give this a full-blown try, using LM399, proper PCB and a little extra effort. Experimenting showed that, in addition to the two stable points magic mentioned, there could be some additional more or less stable points at very low currents, and with parasitic capacitance on the ballast side it could get stuck there (happened on breadboard, where there's more parasitic capacitance), I did put in some JFET that will kick-start the voltage, and then in essence just influences the circuit via gate leakage (I hope).
Overall I went with a pair of 10k low TC ballast resistors in parallel with the LM399, mostly because 10k were obtainable for a reasonable price, and 1.4mA is not that far off the suggested 1mA... Also used matched mirror SOT23 parts for the current mirrors, no ballast, but a pair of L7812 (one for the reference side, one for the heater) to suppress more supply noise.
I did build a total of four of these, though the first one was stillborn (messed up soldering temp...), and the last one plagued by an LM399 specimen with a lot of popcorn noise (this one featured two plateaus separated by about 7µV, very obviously). But the other two turned out fine, and as far as I can tell have not drifted any more than my measurement uncertainty since then (apart from an induced drifting event when I modified one of them as an experiment).
They usually are quite low-noise also (but I never managed to achieve more than 10 minutes of continuous recording without some externally induced spike) - over those 10 minute intervals the Keithley 7510 showed standard deviations between 0.6 and 1.2µV, although there seems to be some temperature dependence overall (as it got cooler in the evenings the voltage changed ever so slightly, somewhere in the single µVs).
Overall these seem to work quite well for my very limited skills...