BJT variable Voltage Gain Distortion
This is not a good circuit for several reasons:

The bias point is unpredictable, since it's quite dependent on the gain of the transistor.  The effective resistance at the base is roughly RE times the gain.  That means that the voltage divider of RB and this apparent base resistance varies strongly with the transistor gain, which in turn makes the bias point unpredictable since the gain of bipolar transistors is unpredictable.  All you usually know is that it will be above some maximum level, not how high it could be.  It's not at all unusual for actual devices to have several times higher gain than their minimum guaranteed gain.

The gain adjustment also effects the bias point.  Even worse, nothing is compensating for this change.

The output impedance, and therefore the output rise time, change with the gain setting.



To fix these things, first create a circuit that has the highest gain you ever want and bias it predictably so that the output is close to the middle of its range with no signal in.  Then put a potentiometer on the input of that amplifier to adjust overall gain.  If the input signal has a DC offset, then AC couple the potentiometer both from the input signal and from the input to the amplifier.  The potentiometer will therefore always operate with a DC level of 0, so changing the gain won't change any DC level anywhere.