With present thermoelectric effects, using a Seebeck junction to generate current for a fan is hopelessly ineffective. But is that necessarily the case for all designs which could help to hold a system under a critical temperature when heat spikes.
Why isn't it also true that you can only make something warmer, by cooling something else by a larger amount?
The movement of electricity generates waste heat, why isn't that process reversible? Making the heat disappear into a cold wire, rather than just dissipating into the atmosphere? (not suggesting it's would be easy or even practical).
[1] https://www.britannica.com/science/Seebeck-effect
[2] https://www.amazon.com/Peltier-Cooler/s?k=Peltier+Cooler