I prefer to use the opamp because I would have stable oscillations.
I found some single supply voltage opamps, like LM386 and LT1006.
Find the attached circuit.
Now, the problem is this.
If the object is near the sensor, the capacitance will increase, maybe to some % of farad, and the frequency of the oscillator will decrease toward a a dc value. If the object i far, in theory the capacitance should be 0. As the object disappear the frequency will increase.
At the limit, it should be infinity, but I think that a dc value will appear.
So, in order to detect the object, I thnik to add a counter of pulses.
Is it good?