I am struggling to understand why eletret mics always (so far as I can see) use an amplifier while dynamic mics usually have a transformer.
Probably because a dynamic mic can produce an appreciable amount of power on its own, so it's feasible to use a transformer to step up the voltage. But the power available from an electret mic is negligible, so you need an amplifier with its own power source.
Another reason for using a transformer might be to protect the microphone's low-impedance coil from being damaged by DC.
Here is a mic that I am thinking of using. The datasheet gives it an output impedance of 680 ohms @ 1000hz
https://www.mouser.co.uk/datasheet/2/670/cmi_5247tf_k-1777142.pdf
That microphone looks like it already has an amplifier of sorts built into it to lower the output impedance.
Now, I can find an audio transformer with a primary impedance that in the same ballpark. Why not use it?
A couple of things to understand here:
First, you shouldn't be thinking in terms of "matching" impedances here, in the sense of making them equal. That's only important when you're dealing with RF and have to worry about reflections in transmission lines.
When connecting a signal source to an audio amplifier, the aim is to get as much signal voltage at the input as you can. To get that, you want the output impedance of the source to me much
lower than the input impedance of the amplifier.
Second, the impedances you see in the specifications of audio output transformers are not the impedance of the transformer! The transformer doesn't have an impedance itself, rather it transforms impedances, in the same way that it transforms voltages and currents -- except that the impedance transformation ratio is the square of the turns ratio.
So if, for example, you see an audio output transformer specified as "800 ohms primary, 8 ohms secondary", all this really means is that it has a turns ratio of 10:1 (i.e. sqrt(800/8)). If you connect a load of 8 ohms to the secondary, you will see an impedance of 800 ohms at the primary. Or if you connect 16 ohms to the secondary, you will see 1600 ohms at the primary, etc.
Now if you want to use this transformer with your mic, you want to step up the voltage, so you'll be using the winding with the smaller number of turns (the "secondary" when used as an audio output transformer) as the primary and connecting it to the microphone. Your 680 ohm mic will then appear as a source of 68k ohms to the amplifier. As long as the amplifier's input impedance is subsantially higher than that, you should get about 10 times more signal than you would from connecting the mic directly to the amp.
How well this would actually work in practice is hard to say, you'd have to try it to find out.
Personally I wouldn't bother trying to use a transfomer; if I needed more signal I'd build a simple preamplifier. It would likely be cheaper and have less detrimental effect on the sound quality.