A few other reasons that power use can be important:
*form factor (lowering cooling requirements can facilitate a lighter, smaller, and/or less exposed system; this is a factor with servers where data center space costs money as well as consumer electronics and deeply embedded systems)
*product cost (larger heat sinks add cost, fans add cost--material (including inventory) and assembly--, heat management adds design cost)
*reliability (keeping temperatures down reduces soft errors and hard failures, in addition power saving techniques like DVFS can also increase MTTF by reducing electromigration et al.; removal of active cooling can remove a point of failure--particularly one with a moving part--, even reducing active cooling requirements can improve resilience; tighter integration facilitated by lower power can also reduce vulnerability to mechanical stresses and external electromagnetic interference)
*performance (when performance is limited by TDP, energy-efficiency can increase performance; in addition, if the number of external connections (pins) for power and ground can be reduced, more pins can be available for signals increasing available signal bandwidth; lower power can also facilitate tighter integration which can improve latency and bandwidth)
There is also a distinction between chemical batteries and other power sources. Energy harvesting techniques and radioisotope power cells have different constraints than chemical batteries.
I realize that including all of the above in the introduction would have added too much length, but it is easy to forget how multifaceted power concerns are.