How does the Colpitts oscillator provide positive feedback?
The feedback path is a third-order lowpass (ladder topology): ro-C1, L-C2.
This assumes that the parallel connection ro||RF is app. identical to ro (Transistor output resistor ro). Furthermore, the impedance of the coupling capacitor C3 is neglected.
The output of this lowpass (node between L and C2) is fed back to the amplifier which provides 180deg phase shift.
There will be one single frequency fo where the feedback circuit (lowpass) also provides 180 deg phase shift (3rd order with max. -270 deg phase shift). This allows a 360 deg phase shift at the oscillator frequency fo.
General answer to your question:
The oscillation condition (Barkhausen) is formulated in the frequency domain (gain and phase of the frequency-dependent loop gain). This condition requires a certain phase shift (360 deg) for one single frequency only. This implies that the ability of a circuit to oscillate must be analyzed in the frequency domain.
Therefore, your attempt to make an analysis in the time domain cannot be succesful.