I don't know how people can convince themselves that they can understand these effects.

Unusually effective heat insulation is exactly how they work. (This is tempered by eg radiative losses, so making them thicker doesn't work better.) placing poorer heat (vs electrical) insulators between peltier material is counterproductive, similar to using resistors to improve conduction between copper wires

Don't ask me for a better explanation :)

As to why COP or even Carnot efficiency hasn't been thrown out in favor of temp-difference independent efficiency metrics like exergy.

I can't tell you either