Thanks for doing the extra measurement with series resistors. So the slightly nonlinear resistance of the H11F1 seems to be not the culprit for the nonlinearity.
As the errors got larger with higher frequency, it might be a good idea to do future tests at a higher frequency. The H11F1 has a specified response time of 15 µs. Assuming at least 3 time constants, this would be about 50 µs or a 20 kHz limit for the clock to the 4017 and thus 2 kHz for the transformer.
For the effect of inter winding capacitance, it should be possible to test this, by adding capacitive loading and watch the change in voltage. Due to the nearly square waveform, I have some doubt that this is the problem. Anyway it might be interesting to know / check the output impedance of the transformer. I would expect something in the 1-10 Ohms range - so capacitors should charge quite fast.
Form the Datron circuit it looks like they use a frequency of slightly less than 1 kHz. But they also use faster switches and I don't think they want absolute accuracy, but just stable values. For just doing the 7 to 10 V step for a 10 V reference, there is no need to have an absolute value, just stable would be good enough. However understanding why the value is different from ideal can help to make it more stable and having something like accurate 1,2,....,10 V steps would be really nice.
Edit:
I just realized one source of error:
The capacitance in off state together with the now 2 µF capacitor make up a voltage divider that reduces the output voltage. The capacitance of the H11F1 is supposed to be in the 15 pF range (at 15 V), but it is expected to get larger at lower voltage. This error component should be independent of frequency and scale with the capacitance. The shape and size of the observed nonlinearity looks reasonably for this effect. One might be able to measure the H11F1 independently and than calculate the effect. The effect might be smaller with other (e.g CMOS) switches, but the principle problem should be the same.