In an astable multivibrator, why doesn't negative feedback prevent the output from ever saturating?
why is the negative feedback not coming into play and quickly equalizing the two voltages? Can we assume that the positive feedback of the schmitt trigger works faster than the negative feedback involving a resistor and a capacitor?

Yes.
The voltage at (-) terminal can rise and fall only at a speed with which the capacitor can charge/discharge. That rate is limited by the opamp saturation voltage and the resistor Rf. It has time constant Rf x C.
The positive feedback through the two resistors R1 and R2 are nearly instantaneous.