On your original ckt, try changing the 100k to 82k.
When Vref is offset from the middle of the swing of the Schmitt-trigger, one of the Schmitt
thresholds can get pushed closer to one of the supply rails.
If it gets pushed too close the integrator doesn't have enough swing to hit the Schmitt threshold
and the oscillator stops.
For example,
Vcc =9V, Vref = 5.2V.
Opamp swing 2V from each rail, ie. 2V to 7V. ; that applies to both the Schmitt and the Integrator.
R1 = 100k and R2=150k
Schmitt trigger thresholds are 7.33V and 4.0V
Since the integrator swing is limited to 7V it cannot reach the 7.33V threshold.
If we change R1=82k then the Schmitt thresholds are 4.2V and 6.95V, just scraping through.
Alternatively if Vref was in the middle of the opamp swing, (2+7)/2 = 4.5V then
with R1 = 100k and R2 = 150k the Schmitt thresholds are 5.87V and 3.13V.
In this case the integrator swing of 2V to 7V will easily hit the thresholds.
When the integrator's Vref is not in the middle of the Schmitt output swing the duty cycle waveform is non-square.
If you want the integrator to swing about a different voltage than mid-rail you can use two Vrefs,
one for the Schmitt and one for the Integrator. This lets you keep the waveform square.
You can see the general idea in the Small Clone, two Vrefs come from the 180k+68k+82k divider. The integrator
swings about 4.09V and the Schmitt thresholds are centered on 2.2V.
https://sites.google.com/a/davidmorrin.com/www/home/trouble/troubleeffects/electro-harmonix-small-cloneAnother interesting aspect is if Spice software can predict this kind of behaviour?
Forgot to mention, yes spice can do all this stuff. However, a lot of spice models do not have the correct output swing. So if you to simulate circuit like this you need to match the model swing with the swing of the actual part. (The swing tabulated in the datasheet often is no help. The internal circuits are some help.) I've got quite a number of LM741 and LM324 opamp models and only a couple of the models have output swings like the real parts.
Something else, and more likely the problem. The LED is hanging off the output of the integrator. The LED pulls a fair bit of current and that will cause the integrator swing to be further away from the +V rail than if there were no LED load.