What's the need to set the amplifier input common mode voltage to mid–supply voltage in this non-inverting microphone pre-amplifier circuit?

 I answered this SE EE question a few days ago. Here is my answer.

How do we explain circuits?

Circuits are truly understood when, in addition to the specific circuit solution, we show the general idea behind it. This allows us to show the relationship between seemingly different circuit implementations and they will be easily understood.

Biasing

What is "biasing" in electronics?

Obviously, voltage devices require their input and output voltages to be within the limits of the supply voltage; so positive-supplied devices need positive input/output voltages and negative-supplied devices need negative voltages. When signals do not meet this requirement (e.g., AC voltages which "wiggle" around ground), figuratively speaking, they must be "moved", "shifted" inside the working area... and this is called "biasing" in electronics.

How do we bias circuits?

The remedy is obvious - we add/subtract "shifting" voltages to/from the input and output voltages: for positive-supplied devices, we add constant voltage to the (AC) input voltage and subtract constant voltage from the output (AC + DC) voltage; for negative-supplied devices, we subtract voltage from the input and add voltage to the output voltage.

Conceptual bias circuit

To realize the bias idea above, we need a voltage summer. We can make it by the help of KVL, simply connecting a constant voltage source in series. The problem is that this source cannot be connected to ground; it is floating since usually the input voltage source is grounded.

Biasing capacitors

If the input signal is AC voltage, we can use charged capacitors as floating voltage sources (historically the first biasing idea). You can figuratively think of them as "rechargeable batteries". We have to keep them continuously charged to Vbias. Let's see how.

Input circuit. First at all, we have to ensure a path for the charging current through the input voltage source; so it has to be conductive ("galvanic"). Then, we have to supply the other end of the capacitor by a "bad" biasing voltage source with relatively high resistance (voltage divider). As a result, the "coupling capacitor" will charge through the input voltage source to Vbias.

Output circuit. The output biasing is similar but here the capacitor charges through the load (so it has to be "galvanic").

OP's arrangement

Now we can see all these bias elements in the specific OP's circuit.

A clever trick here is that the output bias voltage (the op-amp quiescent output voltage) is created by following the input bias voltage (created by the R2-R3 voltage divider).

Another trick is to connect a resistor (R6) in parallel to the load thus making it "galvanic" when needed (e.g. in the case of an open circuit).

C2 shows another capacitor application. While C1 and C4 are "moving" floating capacitors (aka "coupling capacitors") that convey the voltage variations, C2 is a "fixed", "immovable", grounded capacitor (aka "decoupling capacitor") that "freezes" the voltage variations across itself.

See also

Here are links to some of my materials related to the question:
Circuit Idea (how to understand, explain and invent circuits)

What does "biasing" mean and how is it implemented in electronic circuits?

What do coupling capacitors really do in AC amplifiers?