Why BJT BE voltage allows to avoid distortions
In your circuit, the biggest factor in play is R4, the emitter resistor.  This provides negative feedback in the following way.  As the signal rises, the collector current increases with it, and so does the emitter current.  This causes a voltage drop across R4 which very nearly match the rise in base voltage, so V-BE doesn't rise much at all.  There's also a corresponding reduction in gain because of this.  The same thing happens when the signal goes negative, as long as the transistor was originally biased into conduction to start with.

I will also mention that a BJT is a current-operated device.  As you try to raise the base voltage, it draws more current and the voltage doesn't rise by much.  The signal voltage appears instead across the series base resistor...which you don't have. You do have C1, so this effect would be present at low frequencies. In real life, most circuits would have a series base resistor.