I am still grappling with how the current triggers the avalanche rather than reaching and exceeding Vz.
You don't. There is no magic trigger, or early mode. That's why you cannot measure anything unusual.
The prose is a bit creative, claiming some special effect.
See:
https://en.wikipedia.org/wiki/Zener_effect In general, diode junction breakdowns occurring below 5 volts are caused by the Zener effect, whereas breakdowns occurring above 5 volts are caused by the avalanche effect.[3] Breakdowns occurring at voltages close to 5 V are usually caused by some combination of the two effects. They are all called Zener diodes for simplicity, but >>5V are actually avalanche diodes.
At very low (100uA) currents some papers talk about a break-back effect, but that is a low frequency sawtooth, and I've never seen a real diode behave that way, as the VZ would be out of spec.
You can buy 50uA zeners, so that does not seem to be a fundamental issue.
https://www.nexperia.com/products/diodes/family/50-UA-ZENER-DIODES/A number of vendors offer MMSZ4709, 24V tested at 50uA
For an RF noise source, you want to actively avoid any low frequency effects.
When the zener is conducting, it has a quite low impedance, so I doubt it matters to the zener if it is nominally current driven, or fed from a resistor at constant voltage.
A low ripple supply is desirable in both cases.
A common circuit for Zener control is to have the zener 'provide its own voltage'. A an opamp has dual feedback paths for an output some ratio above Vz, and then Vz is fed from that output.
See circuit #1 here
https://electronics.stackexchange.com/questions/283651/opamp-zener-regulator-configurationThat has high PSRR, tho needs checking it starts reliably.
For RF noise extraction, you may also need to watch the natural RC rolloff. eg
BZX58550-C24 Zener with 5mA is 70 Ohms and 55pF , Tau of 3.85ns
PDZ24B at 5mA is 30 Ohms, max 80pF for Tau 2.4ns
ROHM UDZV24B has curves for Cz and impedance, suggests 3pF > 15V and ~ 60 ohms 1-10mA for Tau ~ 0.18ns
With very low Cz zeners, finding a current source that does not impact that would be tricky, so a simple resistor feed would have less impact on the rf noise spectrum.