Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a homology theory. I've also seen many uses of the theorem, to explain features about Eilenberg-MacLane spaces and other (categorical) phenomena which relate homotopy and homology. But,
Is there an intuitive reason (geometrically?) why it's true?
Is the Dold-Thom theorem to be expected? Why would one come to think of this?
It is very intuitive and clear in low degrees, but the geometry might stop after this. The $i=0$ case is settled by connectivitythe connectedness of $X$. The $i=1$ case can be seen fromis the ability to lift and commute loops, when analyzing the compositions $\pi_1(X)\to \pi_1(X)^d\to \pi_1(\text{Sym}^d(X))\to H_1(\text{Sym}^d(X))\to H_1(X)$ by "lifting. Perhaps I can argue similarly in higher degrees when $X$ is a closed Riemann surface. This is clear for the sphere, since $\text{Sym}^d(\mathbb{C}P^1)\approx\mathbb{C}P^d$ and commuting loops"$\pi_i(\mathbb{C}P^\infty)\cong\tilde{H}_i(\mathbb{C}P^1)$. Note that this is also clear in the 1-dimensional case when $X\simeq S^1$, as $\text{Sym}^d(\mathbb{C}-\lbrace0\rbrace)\approx \mathbb{C}^{d-1}\times(\mathbb{C}-\lbrace 0\rbrace)$. The content of the theorem is reduced to low degrees for these simple examples.