The LM399 has quite a few not so good points:
1) The temperature regulation is set to a rather high temperature. A lower temperature could have improved drift and noise. Not many application need such a high temperature due to something like a 75 C board temperature.
2) The zener current is rather low and fixed - so a relatively high noise, but still good enough for a 5 digit DMM.
3) Using the Shunt configuration means that there should not be a significant load current to the reference circuit. This is a little inconvenient, but not that bad. OP with well lower noise and drift than the LM399 are not that special.
4) The shunt configuration is only good for a limited driving power of the reference circuit - due to the internal gain it is better than just the raw zener, but not good to drive a variable current. A true ref amp can drive higher currents without much negative influence.
Still it is a widely used reference - maybe because it is not that expensive. At something like $12-15 it is well affordable, as it does not need expensive resistors in the support circuit. A fixed temperature and fixed zener current also means there are no external precision resistors needed to set the current and temperature. The integrated temperature control is rather convenient, though it means extra power that could be avoidable in principle. For highest performance a separate reference and control circuit (like with the LTZ1000) has some advantages. One is that one could use good size capacitors for the usually relatively slow temperature control.
Using a higher current for the zener is a two sided thing. In a temperature stabilized setup one usually needs a heater power that scales with the power requirements of the reference itself. Something like 2-5 times that power (under normal temperature) seems to a reasonable compromise to have so reserve for a higher room temperature. The 2DW232 zener, if used with the second diode in series is more on the high current side (e.g. 20 mA range, but varying). For a heated reference this is rather on the higher side. If it is just for the noise, a lower current in the 1-2 mA range might be good enough.