Beta reduction capacitor in Sziklai Pair speaker driver
Your simulation is not completely showing the function of this circuit.  The output of IC3a is a running at around 20kHz.  The funtion of 555 and what you are calling Q1 and Q2 is to shift this frequency down to something audible.  Men tend to not be able to hear 20kHz.  You need to add a second signal source through the 22k resistor R25 to see what is happening.  This circuit is actually a crude mixer.  You also need to run your 555 pulses at 6V and the Vcc at 9 to get the intended result.