mystery of emitter follower output impedance

If the Vsig source is a voltage source with zero output impedance, then the impedance at the emitter is re + Rsig/(β+1).

Just having Vin be zero doesn't change that, because if Vin is zero that just means that Vsig must be zero (small signal value, of course). The impedance driving the base is still Rsig even if Vsig is zero.

The only way, for the circuit shown, that the impedance at the emitter (Ro) is going to be just re is if Rsig is zero. The full expression Ro = re + Rsig/(β+1) includes that possibility.