If one worries about all the possible sources of drift, using 1% metals and an IC socket should doom this circuit from the start. OTOH, my experience is once it settles down for a few weeks, it stays within a couple ppm for several years at a time. FWIW, my 731A uses metal films for the zener current and manganin wound resistors for the dividers (I think). The 731A has it's last official cal more than 10 years ago and in comparison with three other reference units, it hasn't moved more than 1-2 ppm. A lot of the drift specifications you see become much better after things have aged a few years. I'd rather have an ancient reference with a lot of history, than a brand new one with zero history!
My "oven" was nothing more than a cardboard box over the board with a lightbulb shining on it. Warm the board up a dozen degrees, cool it down (all slowly) while monitoring the voltage. Change current until you can cycle it with no significant change. If you temperature controlled the circuit, it probably wouldn't be capable of sub-ppm performance, at least not without better resistors. It's very good, but can't break the laws of physics. Or Ohm. IMO, sub-ppm performance is vastly harder to achieve and much more expensive as you need better meters and comparison standards. that's why we have threads on the LTZ parts! This circuit is a great way for somebody to get their feet wet with a discrete design, and for many it's all they might need.
As for decoupling TC and gain, you have to change the zener current to find the zero TC point, and that changes the zener voltage. The zener curve isn't a straight vertical line. Thus, you have to change the gain to get back to 10 volts, and that's just the nature of the thing. Adding more complexity than the minimum shown, isn't going to improve things. Certainly though, you can use this circuit with any sort of two terminal reference device.