I don't claim to have a complete answer but here are some miscellaneous comments.
Note that topological spaces are already very nearly defined to be dual to certain commutative algebra-like structures, namely their frames of open subsets. The simplest interesting case of this duality is a duality between finite sets and finite Boolean rings / algebras. One way to think about this conceptually is that the open subsets of a topological space axiomatize verifiable or semidecidable properties of a point in that space: that is, properties such that if they hold you can check that that's true, but such that if they don't hold you can't necessarily check that that's true. See this math.SE question for more discussion on this point. For example, the open sets in the product topology on $\{ 0, 1 \}^{\mathbb{N}}$ correspond precisely to those properties of an infinite sequence of zeroes and ones that you can verify by looking at finitely many terms of the sequence. (Once we agree that verifiable properties are a cool thing to look at, we should also agree that we care about logical operations on them, like AND and OR, and this is where the commutative algebra-like structure comes from.)
One way to define the category of commutative $k$-algebras is as follows. Let $\text{Poly}(k)$ be the category whose objects can be thought of as the affine spaces $\mathbb{A}^n$ over $k$ and whose morphisms $\mathbb{A}^n \to \mathbb{A}^m$ are $m$-tuples of polynomials in $n$ variables over $k$, with composition given by composition of polynomials. On the one hand, this is a Lawvere theory, and the category of commutative algebras over $k$ can be defined as the category of models of it, or more explicitly as the category of product-preserving functors $\text{Poly}(k) \to \text{Set}$. On the other hand, this is a full subcategory of the category of varieties over $k$, and one can try to probe varieties by mapping affine spaces into them; this turns a variety into a presheaf $\text{Poly}(k)^{op} \to \text{Set}$. There is a general relationship between functors and presheaves on the same category called Isbell duality, and the nLab suggests that this is the general setting for adjunctions between things that look like spaces and things that look like commutative algebras, although I haven't really internalized this.
The two points made above can be related as follows. One way to think about the category of sets is that it is the category of ind-objects of the category of finite sets; equivalently, it's the category of presheaves $\text{FinSet}^{op} \to \text{Set}$ sending finite colimits to finite limits. Now, $\text{FinSet}^{op}$ is the category of finite Boolean rings, and the category of Boolean rings can be thought of as the category of ind-objects in this; equivalently, it's the category of functors $\text{FinSet} \to \text{Set}$ preserving finite limits. These two descriptions should be related by Isbell duality.
Why is $\text{FinSet}$ so important, anyway? Well, one categorical property that categories of spaces often share that isn't true in general is that binary products tend to distribute over binary coproducts. This is true in particular in any cartesian closed category, such as $\text{Set}$, and any category with this property behaves in some sense like a categorified commutative ring (!). It turns out that $\text{FinSet}$ is the free distibutive category on a point.
