A $20 cooler gives you the full warrantied performance of the 13900k ,I don't think that you can go much cheaper than that on cooling for anything...
The Core i9-13900K can push even the most capable liquid coolers to their limits, but you don’t lose that much performance if you go with budget-priced air cooling.
www.tomshardware.com
Also with the same amount of cooling the 13900k is cooler than at least the 7950x at any power level, sadly no info on the 3xd they might be better.
www.anandtech.com
Only if you plan to use your PC without an GPU, and even then you might win about 100W if you overclock the intel system to the limit compared to the stock ryzen, it will be about 200W for ryzen compared to 300-350 for overclocked intel.
This will be the same level of PSU for both, nobody is going to use a 200W PSU on an expensive CPU like that, even 450-500W would be a huge stretch.
You save a bit on the electricity bill but then you can also just set the power limit for the intel to the warrantied limit of 253W or even lower.
As the first link above shows the difference in performance between a $20 ( around 240W) and a $200 cooler (315W) is about 6%