Homology theory constructed in a homotopy-invariant way

Throughout the following, I'll say "homotopy category" when I really mean the weak homotopy category.

For a space $X$, the homology of $X$ is canonically isomorphic to the reduced homology of $X_+$, which is $X$ with a disjoint basepoint. Therefore, it suffices to give a definition of the reduced homology of a based space.

The smash product $\wedge$ descends to a well-defined operation on the homotopy category of based topological spaces. For any $n \geq 0$, we have an object $K(\mathbb{Z},n)$ in the homotopy category, and weak equivalences $K(\mathbb{Z},n) \to \Omega K(\mathbb{Z},n+1)$ which are adjoint to maps $S^1 \wedge K(\mathbb{Z},n) \to K(\mathbb{Z},n+1)$. For any integer $m$, we can therefore define a directed system of sets $$ [S^{m+k}, K(\mathbb{Z},k) \wedge X ] \to [S^{m+k + 1}, K(\mathbb{Z},k+1) \wedge X ] \to \cdots $$ The colimit $colim_k \pi_{m+k}( K(\mathbb{Z},k) \wedge X)$ is isomorphic to the $m$'th reduced homology group of $X$ in a canonical way.

This kind of definition produces generalized homology theories, and this all falls into the subject of stable homotopy theory.

The method you suggest of taking the free spectrum in this $\infty$-categorical sense will give a homology theory, but instead of producing the homology groups of $X$ it will produce the stable homotopy groups. The "free abelian topological group" on the topological space $X$ can be used instead, and shown to give a good notion on the homotopy category; this will produce the homology of $X$, as a result of the work of Dold and Thom.

I don't know what you are looking for, but here is my idea:

You could use the universal property of the homotopy category among "higher categories" as follows: For any "higher category" $C$ and any object $c$ of $C$, there is up to isomorphism one functor from the homotopy category of spaces - let's call it $H$ - to $C$ sending the point to $c$. Here, "higher categories" could mean homotopy categories of model categories, where the functors we allow are derived left Quillen functors, or (I guess) $\infty$-categories.

The derived category of $\mathbb Z$ is such a category, so we find a unique functor $F: H \rightarrow D(\mathbb Z)$ sending the point to the chain complex $U$ which has only $\mathbb Z$ in degree $0$. Now $D(\mathbb Z)$ is triangulated and we can define $H_{\star}(X) = [U, F(X)]_*$ for $X \in H$. It is easy to check that this is a homology theory and has the correct behaviour on the point. (This approach also has the advantage that the Künneth formula follows from abstract nonsense.)

To me, something like this seems to be the only way to avoid the mentioning of topological spaces or simplicial set, since the category $H$ either has to be given explicitly or characterised by some property as above. And introducing "higher category" structures also seems natural since you cannot properly define a homology theory on some random category; you need some additional structure such as cofiber and fiber sequences and suspension etc.

There is also the notion of an algebraic model category (and probably there is the same notion for $\infty$-categories), and our functor $F$ is the initial functor from $H$ to the homotopy category of an algebraic model category. In this way, one could view singular homology as the universal algebraic invariant of a homotopy type, but this is rather vague.