Shure discusses a power supply circuit "... designed to operate an electret condenser microphone." It may be a bit of over-kill, but you can see from this schematic that they are very serious about the idea of filtering the supply rail:
simulate this circuit– Schematic created using CircuitLab
They also write a footnote: "Important: The circuit above must be in a metal (shielded) enclosure." That note applies to a box with XLR jacks that may be attached to cables going to the electret microphone. But you may note just how important it is to protect signals that may be in the \$\mu\text{V}\$ range.Part of your circuit may be carrying such small signalling, prior to amplification. A solderless breadboard with long wires and parasitic capacitance is a one problem. And their suggestion for the electret supply is also already a problem. That's two problems.
There are at least two more problems.
Emitter follower speaker drivers have only one active quadrant, the other being passive; they add unwanted distortion no matter how you design them; and to get them to sound tolerable they must be designed to dissipate very heavily. After going to all the trouble to remove noise, taking Shure's recommendations about supply filtering and shielding seriously, it seems shameful to then recommend an emitter follower driver to operate the power stage driving a speaker.
The opamp voltage gain showing in the schematic is \$\vert A_v\vert=11\$. I've no idea which electret you are using, but adafruit offers a CMA-4544PF-W electret condenser microphone with this datasheet. A relatively loud conversation (as you'd perceive it) would reach the microphone at about \$60\:{\text{dB}_{\small{SPL}}}\$. Assuming the datasheet conditions of operation, yielding about \$125\:\mu\text{V}\:\pm 25\:\mu\text{V}\$. Amplified as indicated in that circuit, and assuming no attenuation at the voltage divider (which will attenuate), the output will be \$1.40\:\text{mV}\:\pm 300\:\mu\text{V}\$ at the speaker. This will be about \$125\:\text{nW}\$ at the speaker. (Yes, that's nanoWatts!)
Even with the hard-to-find Panasonic WM-61A, which is a very good electret, the speaker output would be about \$1\:\mu\text{W}\$.
A final point.
Although Analog's circuit doesn't provide any filtering at all, let's compare Shure's arrangement above with a simple RC filter where the \$R=1\:\text{k}\Omega\$ and the \$C=47\:\mu\text{F}\$(which is infinitely more than Analog suggests):
At \$100\:\text{Hz}\$, Shure's filter is down about \$-71\:\text{dB}\$. The simple RC filter is down about \$-29\:\text{dB}\$. Suppose there's only about \$1\:\text{mV}\$ of \$100\:\text{Hz}\$ ripple on the power supply. With Shure's arrangement this means about \$300\:\text{nV}\$ at the electret supply rail. With the simple RC, this means about \$35\:\mu\text{V}\$ at the electret supply rail. From this fact alone, and knowing that that a loud signal might produce only about \$125\:\mu\text{V}\$, it seems obvious that Shure's extra measures on the supply rail may very well be worth the extra effort.
Shure knows what they are doing.
Yes, electrets are finicky.
I've no good idea where their minds were at on that web page, with long exposed wiring on a solderless breadboard, low voltage gain, zero electret supply rail filtering, and a crappy emitter follower speaker driver.
It is no surprise that "the volume is extremely low" and that the "audio coming from the speaker is very noisy."
What you found is the expected result from that circuit from Analog's circuit and construction.