I don't understand, what am I missing? The heat pump increases efficiency by having COP 2-4 right? Assuming air to air and being in, say, Denmark.

Heat (above 100C, say, burning garbage) to electricity: 50% (theoretical best case)

Electricity to heat (around 40C): 200%-400%

Net win?

The surplus energy comes from air or ground temperatures..

Yes you cannot heat back to the temperature you started with but for underfloor heating 40C is plenty. And you can get COP 2 up to shower water of 60C as well.