I generally see the class equation stated for the action of a group on itself by conjugation as follows where $C(G)$ is the centralizer of $G$:
$$|G|=|C(G)|+\sum{sizes\ of\ nontrivial\ conjugacy\ classes}$$
...or...
$$|G|=|C(G)|+\sum [G:C_G(x)]$$
It seems pretty clear that the more general statement also true. That is, if $G$ acts on a set $S$:
$$|S|=|Fix_G(S)|+\sum{sizes\ of\ nontrivial\ orbits}$$
...or...
$$|S|=|Fix_G(S)|+\sum [G:Stab_G(x)]$$
Pardon the poor notation. In both cases, I am summing over non-trivial conjugacy classes, and $Fix_G(S)$ indicates the set of points in $S$ fixed by all $g\in G$.
First, I am asking if my overall statement is correct.
Second, if the overall statement is correct, I was wondering if there is a reason why the conjugacy class statement seems to get more prominence than the general statement. Is the general statement not very useful?
Yes, the orbits of a group action partition the space that the group is acting on. If that space is finite, then its cardinality is the sum of the cardinalities of the orbits. Moreover, you can indeed apply the orbit-stabilizer theorem to re-write the equation the way that you have.
The class equation is just this observation applied to the action of G on itself by conjugation. Partitioning a group into conjugacy classes is a major feature of group theory, so it's helpful to have a name like "the class equation" to quickly identify what you're talking about.