I connect many of these together and eventually I have 1.1 W from my 1 W input. I can keep going in this manner, at some expense sure, but now I’m producing 2 W per 1 W input. The effect must be cumulative if the device is truly a perpetual motion machine.
It will cost me less to mine resources giving me economic advantage. Eventually as I build more energy production capacity it will cost me nothing. I can harvest as much raw material as I need and fabricate whatever machines I want at zero cost. Currently this only limited by wealth but in my free energy utopia I’m unconstrained except by availability of material resources, I have infinite energy.
So I build a lumbering ship unconstrained by the limitations of mere rockets and mine the planets. I can build a planetary civilisation from this. I have enough energy, mechanical and computational power to do this easily. I now need to devise a means to achieve interstellar travel. But I have infinite energy, therefore infinite compute and near limitless material resources across the solar system.
Because of the promise of perpetual motion and free energy the effect snowballs at every step.
Reductio ad absurdum