The pullup resistor is necessary if the servo op amp's output can reach ground because that will also be a stable operating point where its inverting and non-inverting terminals will be at the same voltage. In the ADR1000 datasheet they actually mention that the op amp in their eval board (the ADA4084) can't get its output all the way to ground, so it will start up. I believe the same is true of the LT1013, so it won't actually be able to maintain that stable operating point at 0 V. As Andreas said, the diode is there to prevent forward biasing the Zener. Forward biasing tends to shift the breakdown voltage of Zener diodes, so if it were allowed to forward bias that would be detrimental to stability.
A common complaint I have seen about this circuit is that the residual temperature coefficient (i.e., with the oven off) is not insignificant, which is to say that the zero tempco point (as the temperature coefficient is a parabola with negative curvature) is too high to be useful. I am sure there are people who would know better than me what that residual tempco is around ambient temperature, but I think it's in the range of +20 to +50 ppm/K. Note that when the heater is powered, the chip will only move a small fraction of the ambient temperature change, so the tempco will appear linear as others have mentioned. That the residual tempco is a parabola is most useful when it comes to selecting a heater temperature - ideally it will be near zero around the heater setpoint. I don't know if the tempco of the zener itself is parabolic or if the curvature comes mainly from the compensating BJT, which will have a tempco curvature of about -1 uV/K^2 (
https://web.mit.edu/klund/www/Dphysics.pdf). Another way of compensating this is to add a resistor at the cathode of the Zener, which was done in the Wavetek 7000, which operates at a lower temperature set point than the datasheet circuit. The collector resistors also influence tempco as increasing the collector current of a BJT makes the Vbe tempco less negative. The ADR1000 residual temperature coefficient is closer to zero than that of the LTZ1000, so if you pop one of those in it will probably push the temperature coefficient negative.
One of the reasons that people don't really like that the residual tempco is high is that if there is a residual tempco, the temperature setting divider R4/R5 needs to be more stable with respect to temperature. There aren't any really good 13k/1k or 12k/1k dividers in stock, if you want a BMF one, for instance, you are stuck waiting whatever the lead time is and possibly dealing with MOQs. Discrete BMF resistors just aren't as good, even though the datasheet typical specs suggest they should be. So a clever strategy of compensating the residual tempco loosens the requirements around this divider. An advantage of the cathode resistor strategy here is that you can characterize the tempco over a range and select a heater setpoint where it is minimal, but the resistor is itself influential, which is a tradeoff one must weigh in their own design. Others have incorporated a source follower JFET to provide the zener current, which has the advantage of reducing the current draw (and heating) of the servo amp. The CERN DVM circuit does this, and I have used it in all the references I have made.
To design your own doesn't require knowing the device physics - the things you are looking out for are much more mundane. You don't want heater current flowing in the same path as your sense- voltage, thermocouples with the kovar leads are important, so efforts must be taken to keep them at the same temperature (as mentioned in the datasheet), and be aware of the thermal gradients caused by the reference itself and the heater transistor to avoid unwanted effects from parasitic thermocouples.
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