Intuitive explanation of Burnside's Lemma

I'm not sure I'd call this a categorification, but the way I think of Burnside's Lemma is as follows.

Consider the subset $Z \subset G \times X$ consisting of pairs $(g,x)$ such that $g\cdot x =x$, where by $\cdot$ I just mean the action of $G$ on $X$.

The cartesian product $G \times X$ comes with the two surjections $\pi_G : G \times X \to G$ and $\pi_X : G \times X \to X$, and you can compute the cardinality of $Z$ either along the fibres of $\pi_G$ or along the fibres of $\pi_X$: the former gives you the sum over the fixed point sets, whereas the latter gives you a sum over the stabilizers. Then the orbit-stabilizer theorem does the rest.

Thanks to @Arrow who pointed out the link in my comment was broken. Here's hopefully a link that works to the same one-page document.

One can view Burnside's lemma as a special case of the mean ergodic theorem, which links time averages to spatial averages, which may qualify as "equating two objects of the same type". On the other hand, the mean ergodic theorem is more complicated than Burnside's lemma, so this may not qualify as an intuitive explanation.

Nevertheless: given a measure-preserving action of an amenable group $G$ on a space $X$, the mean ergodic theorem tells us that

$$ {\bf E}_{g \in G} \langle T_g f, f \rangle_{L^2(X)} = \| \pi(f) \|_{L^2(X)}^2,$$

where $\pi(f)$ is the orthogonal projection of $f$ to the $G$-invariant functions, and $T_g f(x) := f(g^{-1} x)$, and ${\bf E}_{g \in G}$ is a mean on $G$.

If one applies this to the one-sided action $g: (x,y) \to (gx,y)$ on the product space $X \times X$ equipped with counting measure, with $f$ equal to the Kronecker delta function $f(x,y) = \delta_{x,y}$, $\pi(f)$ is equal to $1/|O|$ on the square $O \times O$ of each orbit $O$, and so one obtains

$$ {\bf E}_{g \in G} |X^g| = |X/G|$$

which is Burnside's lemma.

Some thoughts. $X$ defines a representation $V = \mathbb{C}^X$ of $G$ with character $\chi(g) = \text{Fix}(g)$, and the projection from $V$ to its invariant subspace is $\frac{1}{|G|} \sum_{g \in G} g$, so the trace of the projection (which is the dimension of its image) is $\frac{1}{|G|} \sum_{g \in G} \chi(g)$. On the other hand, the invariant subspace of $\mathbb{C}^X$ is spanned by sums over orbits, so its dimension is the number of orbits. Phrased this way Burnside's lemma can be thought of as a "trace formula" relating a "geometric" quantity (the number of orbits) to a "spectral" quantity (the sum of fixed points). The value of other stronger results of this kind is precisely that the objects on both sides are not of the same kind, so perhaps it's not natural to expect them to be any more closely related than that.

(I tried a categorification in $G\text{-Set}$ but it didn't lead anywhere interesting.)