Connections between Algebraic Topology and Set Theory (Co) Homology functors are dependent on the homotopy type of the objects they act on and so a lot of results only care about the "loose" classification of spaces (including the use of co-final spectra in showing Adam's stable category is a model for a stable homotopy category as in Margoli's Spectra and the Steenrod Algebra Chapter 2). Similarly, a lot of results about cardinals seem to only depend on the cofinality of the cardinal.
From my limited exposure to stable homotopy theory and set theory it seems like there must be some connection similar to the connection between Galois extensions and covering spaces.
 A: This a comment taken from /u/ultrafilters on reddit.com/r/math which I I think is a sufficient answer.

Generally cofinal-ness is something that appears all over the place in math, oftentimes going by the name of 'dense'. Being able to describe, or cover, a mathematical object in some minimal way can be a good first step in analyzing that object; structural properties of the smaller dense set may be easier to describe and have meaningful implications about the original object. I don't think there's anything uniquely happening in the situation you described. It just happens that density on well-founded orders (like one side of a chain) correspond exactly to density on ordinals.
The further question about "why the 'structure' of ordinals is so closely tied to their cofinality" probably doesn't have a much better answer than saying it gives us a minimal way to cover that ordinal. There are slightly better things to say about the combinatorics of ordinals as related to their cofinality using more definitions, but nothing absolutely satisfying.

