oscillator frequency stability

Does the frequency stability matter in VCO if the tuning range is wide enough to compensate the center frequency drifting over time and temperature. I mean the use of VCO in PLL as a synthesizer

If the VCO frequency is drifting more, the Phase Noise it will increase, because the PLL is trying more to compensate for this drift.

Yes, the PLL will compensate for that drift and zero it out.

For the drift to turn into phase noise, it would have to be a whole lot of drift--more like FM modulation than the type of drift you are probably thinking about.