This is a great question, because the important book by Judea Pearl ("Causality", 2nd ed.) actually calls
$P(Y|do(X = x)) = P(Y(x))$
the "causal effect" of $X$ on $Y$ (Definition 3.2.1), so that
$E[Y(x)] = \sum_y y P(Y(x) = y)$
would be the "average causal effect".
Although the general focus on the ATE as a difference makes sense, there are a few good reasons to focus on $E[Y(x)]$. First, the ATE can always be constructed from $E[Y(x)]$ for varying $x$, but not the other way around. Second, $E[Y(x)]$ is indeed the average outcome after one has implemented $X = x$ in the whole population, which in itself is interesting to know.
I personally would call $E[Y(x)]$ the "post-intervention mean", just like Pearl also sometimes calls $P(Y(x))$ the "post-intervention distribution".