Of course stress is manifest in a variety of ways, not just increased f0, and f0 indicates much more than just stress - but is it accurate that stress always entails at least increased f0?
No, on numerous levels. On a token level, sometimes a stressed syllable doesn't have higher F0 and could have a lower F0. More systematically, in some dialects of English there are intonational pitch overlays on word stress whereby stressed syllables are actually lower than unstressed syllables. You might conjecture that looking just at the robust patterns of citation intonation, stressed syllables are never systematically lower in pitch than unstressed syllables in "true stress" languages (that is, excluding so-called pitch accent languages). The reason for the exclusion of "pitch accent" is that such languages tend to have pitch-peak shift that can move the F0 peak to the edges of a syllable (leading eventually to stress shift), and that can result in an adjacent syllable actually being higher in pitch that the stressed syllable. I am not aware of any language where stress is systematically realized with lower pitch. Finally, there are languages (Makonde, North Saami) where stress has no correlation with pitch, instead it correlates with duration.
As for relating the different manifestations of stress to a phonological representation, it naturally depends on what you take the representation to be. Ever since getting free of the ides of stress being a feature of a segment, people have assumed that stress is some kind of annotation on a syllable that says "this syllable is special", for instance "has more stars piled on it" (is more prominent). Probably because of the diversity of phonetic realizations of stress across languages, phonologists simply stopped at the point of saying "now it's identified, something else spells exactly out how that is pronounced".
That said, one of the most common correlates of stress, syllable weight, is in fact quite phonological and potentially contrastive. Still, representations don't explain why stressed syllables tend to be longer rather than shorter. The observation is reified in various observations (stress-to-weight, weight-to-stress), but I think true explanation has to come from a better understanding of phonetic parsing.
There is a second quasi-phonologized representation of stress diversity, indeed at the root of the metrical theory, namely Liberman's account of melodies, where he proposes that a connection between intonation and lexical stress via tonal melodies. In his system, most languages would turn out to be somewhat pitch-accenty, it's just that the melodic content is not grammatically or lexically determined.