How would a transistor's hFE affect a phase shift oscillator?
The classic 3 section phase shift oscillator requires a voltage gain of 29 to oscillate continually. Of course you do not want continual oscillations since you need a damped sine wave out.

The Hfe of the transistor does not directly affect the voltage gain but will affect the input impedance which will affect the loss in the phase shift network and thus the Q of the resultant circuit.

Your circuit is slightly unusual in that the emitter resistor is not bypassed which should introduce negative feedback (emitter degeneration) and stabilize the circuit with respect to Hfe variations.

I'm surprised that R4 and R5 return to the emitter and not to ground - it's not immediately obvious to me what effect that has.