Sorry, people. The conjecture is false. For a counterexample, take $H=k\left[x\right]$ (the usual polynomial algebra with shuffle comultiplication) and $s=2$. The image of $x^4$ under $\log^{\ast}\left(\exp^{\ast}_2\left(\mathrm{id}-e\right)\right)$ will be $-2x^4$, and this is not in the second term of the coradical filtration.
I was fooled by the fact that (in general, even if $H$ is not cocommutative!) the map $\log^{\ast}\left(\exp^{\ast}_s\left(\mathrm{id}-e\right)\right)$ restricts to the identity on $C^s$. This fact, though, is a triviality, since $C^s$ is a subcoalgebra of $C$ on which $\exp^{\ast}$ and $\exp^{\ast}_s$ are one and the same thing. (This argument doesn't always work over a ring, but this isn't hard to repair. Instead of restricting maps to $C^s$, consider maps modulo maps which vanish on $C^s$.)
A projection from a vector space $V$ onto a subspace $U$ is characterized by two properties: that it restricts to the identity on $U$, and that it maps everything onto $U$. The first of these properties being satisfied, I got overly optimistic that the second one couldn't be that much false. Turned out that it is. (Also, I got overly optimistic by the fact that the conjecture holds for $s=2$ on the third graded component of the tensor Hopf algebra; but the fourth graded component gives a contradiction.)
I am left wondering whether there are some reasonable algebraically-defined projections onto $C^s$ for $s\neq 0,1,\infty$. We shouldn't necessarily look for variations on the Eulerian idempotent; there might also be something similar to the Dynkin idempotent (or the Klyacho Klyachko one, but I haven't even started to learn about that).
