The new design using a single LT1021-10 with a single resistor divider appears to work fine. I was able to salvage the LT1021, TLV170 opamp, inductor and a couple capacitors from the previous board. All of the resistors are new. 

The heater feedback loop works fine. I don't have the Zener diode to limit the swing of the opamp yet, but the loop settles to its steady state value in about 30 minutes. After several attempts to trim the heated 10V reference to exactly 10.00000V I gave up. Here are my initial readings with my now 3 year-old Keysight 34461B DMM:

New Voltage Reference (>3.5hr soak, 62F ambient, 100PLC, HighZ input)

   Reading              Scale            Other                                                     Error

   99.9906mV        100mV          Average of 20 samples, zero null         -9.4uV

   1.0000833V        1V                 Average of 20 samples                         +83.3uV

   9.999987V          10V              Average of 20 samples                         -13uV

So you can see that I was able to trim the LT1021-10 to within 100uV of 10V (according to my 34461B). It's a bit disconcerting that the 1V reference is a bit to the high side while the 100mV reference is a bit low. My initial reaction is that the 34461B is drifting on the 1V scale. If I can get my hands on a calibrated 6-8 digit DMM I will know the truth.

I will keep the reference powered for 1000 hours and trim any resulting drift. After that...time will tell.