Darwin's theory of evolution emphasized that positive selection of functional proficiency provides the fitness that ultimately determines the structure of life, a view that has dominated biochemical thinking of enzymes as perfectly optimized for their specific functions. The 20th-century modern synthesis, structural biology, and the central dogma explained the machinery of evolution, and nearly neutral theory explained how selection competes with random fixation dynamics that produce molecular clocks essential e.g. for dating evolutionary histories. However, quantitative proteomics revealed that selection pressures not relating to optimal function play much larger roles than previously thought, acting perhaps most importantly via protein expression levels. This paper first summarizes recent progress in the 21st century toward recovering this universal selection pressure. Then, the paper argues that proteome cost minimization is the dominant, underlying ‘non-function’ selection pressure controlling most of the evolution of already functionally adapted living systems. A theory of proteome cost minimization is described and argued to have consequences for understanding evolutionary trade-offs, aging, cancer, and neurodegenerative protein-misfolding diseases.