It is true that considering the archimedean places as well is more general, but that still doesn't explain why it is more natural. If we consider both the definitions of an absolute value and that of a discrete valuation, they both seem arbitrary. However, discrete valuations have the extra appeal of being related to geometric objects (they arise as local rings at non-singular codimension 1 subschemes of integral schemes, and in various other situations: local rings at non-singular codimension 1 subschemes of a scheme birational to the original scheme, or more abstract situations like Zariski-Riemann spaces). It seems (to me at least) that the archimedean places don't enjoy a similar geometric analogy.

This leads me to ask: why are the archimedean places natural? Why do they always feel like they're "missing points" of rings of integers?

I should note in the interest of full disclosure that I'm unfamiliar with the ways of Arakelov theory.