C6 should be C0G or film, not X7R. I'd add bigger bulk cap in parallel with C5 (e.g. 100-1000uF 25V/35V).
Also some smaller bulk caps at outputs of LDOs might be nice to have (22-100uF). Check LDO datasheet for recommended/max capacitance.
You need to route heater ground separate from signal ground. Also you should modify schematics to reflect star mecca ground for all signal nodes, this is important. Your connection of zener cathode to heater pin suggests that you missed this important point.
I'd get rid of switch (as unreliable contributor in signal path) and instead have multiple individual output taps (with opamp buffers if you have budget).
Why AD706 chosen?
Also consider new ADR1399, it has noise similar to that of LTZ1000 at fraction of cost/complexity.
There are no 0.2ppm/K resistors, suggest to specify real spec, not typical specmanship

.
Flask is decent idea. Make sure to protect circuit well from air drafts on both sides.
After a year of ageing with constant power and careful calibration and some history data logging you'd be able to use such reference to calibrate any 7.5-digit DMM on fullscale for base 10V and maybe lower ranges. Stability and performance of your divider chain R6-R10 would be biggest error contributor in this design. You might want to consider specially designed resistor network, likes the ones Caddock makes.