Oscillators work because the output gets fed back to the input with just the right phase shift to be the same phase coming out again.
In the case of the that circuit, that means that the output is -- roughly, because the chip has delay -- in phase with the input. It's close to being in your "acts like a resistor" mode.
The weird thing about that circuit is that if you replace the crystal with a capacitor, it will oscillate. In fact, you need the case capacitance of the crystal to make it start oscillating.
So, it starts up oscillating at a frequency determined by the crystal case capacitance and the RC network on the negative input -- basically, it's a relaxation oscillator. The crystal case capacitance working against the pair of 2k-ohm resistors provides the positive feedback that it needs to oscillate.
As it starts to oscillate in that mode, the crystal will start to vibrate. Initially it doesn't vibrate much, but even a slight signal on the crystal will cause the relaxation oscillator to have a component of its output that is at the crystal frequency. The crystal will select for this and vibrate harder, which will have a stronger affect on the oscillation, etc.
I simulated this in LTSpice for a variety of values of the 68nF cap, and even when I had really stupid-low values (220pF, for instance), the thing would still eventually settle out to the correct frequency. With these stupid-low values I saw quite a long period of chaotic behavior on the output, with a frequency of oscillation that was much higher than the "design" frequency -- I suspect that with a real crystal such a circuit would end up oscillating at one of the crystal's overtones, rather than the correct frequency.
Here's the crystal model I used; because I used nice round numbers for the capacitance and inductance, it has a series-resonant frequency of 995kHz instead of 1MHz.
simulate this circuit – Schematic created using CircuitLab