Not sure whether this is an answer to the question but thought of sharing what I feel makes the idea of barycentric subdivision very natural.
On metric spaces one has the crutch of the notion of distance to make sense of what is "small" and then things like Lebesgue Number help one get covers with as small open sets as possible. This notion of smallness gets hard to emulate on general topological spaces and here barycentric subdivision helps get across that by exploiting together the benefit of having a notion of smallness in Euclidean space and the notion of continuity. If something is small in the euclidean norm (as available on the simplex) it will map to something small in an arbitrary topological space under a continuous map.
Further barycentric subdivision sort of very optimally exploits the notion of convexity unlike say thinking in terms of cubes. I am not sure how to make it precise but using cubes instead of tetrahedrons in 3D will bring in many more maps than necessary.
Homology in some sense depends on lower amount of information than a priori it seems to need. Like instead of all continuous maps from the simplex to the space if one takes only those maps which are non-degenerate on some face of the simplex, even then one gets the same homology theory. (This is what Massey does). This kind of think might get harder to see with out barycentric subdivision.
I am not sure what sense it would make if one needed to do infinite barycentric subdivision since in that "limit" one will land up looking at maps from 0 volume subsets of euclidean space to the topological space and won't these simply not allow non-trivial continuous maps from them? I am not sure how to make it precise.
I guess there should be an argument to say that only a finite number of barycentric subdivisions should be enough for any chosen open cover of a "reasonable" topological spaces. I haven't seen a proof of something along these lines in either Hatcher or Massey's book. Would be helpful someone can illuminate this point.
Also the notion of barycenter coincides with the idea of "center of mass" if one can attach equal masses to all the vertices of the simplex. This gives quite an intuitive help. I guess should be cases where this identification is more tangibly fruitful.