Why doesn’t the microphone amplifier work

The noise is very unlikely to be coming from any of the components in the circuit. The biggest culprit is likely to be the power supply. Add a 22uF and 0.1 uF (helps with high frequencies) capacitor and a 100 ohm resistor that feed the electret part of the circuit and a 22uF across the 5V supply.

There are a number of problems with this circuit:

1. The input to the reference is not filtered.

   

   R5 and R4 make half the supply voltage, but also transfer half of whatever noise is on the supply. There should be cap to ground across R4. I'd start with around 2 µF.

2. There is no bias supply for the microphone.

   

   You said this was a electret, so there should be a pullup to 5 V. Usually these should be a few kΩ. Check the datasheet of your mic.

   You should also filter the pullup voltage to keep power supply noise from being fed directly into the mic. Perhaps 1 kΩ from 5 V followed by 20 µF to ground, then 2 kΩ to the mic. Again, see the mic datasheet.

3. There are no bypass caps anywhere!

   

   The above shows the power to only one opamp, but all of them have the same problem. For even better noise rejection, especially for the first stage amp shown above, put a ferrite chip inductor in series followed by 10 µF or so ceramic to ground. That reduces high frequencies, like from radio station pickups, that the active circuitry in the opamp can't handle well.

4. Low gain. The first stage amplifies by a bit over 6. That by itself is OK, although I'd like to see a bit more from the first stage if the opamp can handle it. You didn't provide a link to the opamp datasheet, so I didn't look it up. In any case, there are certainly opamps available that can run from 5 V and provide a closed loop gain of 30 or so at 20 kHz with plenty of gain headroom for the feedback to do its job.

   A microphone amp usually needs a gain of around 1000 at full volume to get to line level signals. Two gain stages at front with a gain of 30 each is usually about right, with a volume control between the two of them.

   After the signal has been amplified, any noise added is a much lower fraction of the whole.

5. Decouple the Vref inputs to each stage. U5 isn't perfect. It will have some noise. It's output impedance also isn't 0, so there will be some cross talk between stages. Even just 1 kΩ in series followed by 20 µF to ground at each point of use would help.

6. Do you really need all that filtering? This is way excessive for any normal audio. Eliminating filter stages will reduce noise. Everything adds noise, so don't put more stuff in the signal path than you really need.