(*) So when $G$ is a topological group, then $[X,G]$ is a group (we can weaken $G$ to a $H$-group if we only consider pointed maps and $X$ is pointed)

Later we can compute singular cohomology for $X$ a CW-complex and $G$ abelian by $$H^n(X;G)\cong [X,K(G,n)]$$

One can go through and show that for $[\Bbb S^1,K(\Bbb Z,1)]\cong [\Bbb S^1,\Bbb S^1]\cong \Bbb Z\cong H^1(\Bbb S;\Bbb Z)$.

Is (*) helpful for computing cohomology in the way of the rest of my message, or otherwise how does one make use of (*)?

@AlessandroCodenotti there's a way to define chain homotopies that basically closely models the topological situation: you can define an "interval chain complex" given by $I=(0 \to 0 \to \dots \to 0 \to \Bbb Z \xrightarrow{(\mathrm{id},\mathrm{-id})} \Bbb Z \oplus \Bbb Z)$ and you have to different "inlusions" of "endpoints" given by taking the "point" chain complex $pt = (0 \to 0 \to \dots \to 0 \to \Bbb Z)$ and including it into the two components of $\Bbb Z \oplus \Bbb Z$
Then if you have a chain homotopy $h$ from $f$ to $g$, where $f,g: A_\bullet \to B_\bullet$, that's the same as a cha…