Of course, you can use a S&K filöter as a core of an oscillator!
However, you need a filter that has a phase shift of -180 deg (together with an inverting amplifier) or of 0 deg (together with a non-inv. amplifier) - in both cases, for a single frequency only!
In the first case, you need a 3rd order lowpass or highpass and in the 2nd case you need a bandpass.
These requirements follow from the oscillation condition (Barkhausen).
Based on a 2nd-order lowpass it will not be possible to realize such an oscillator.