Does a $\mathrm{C}^*$-algebra generated by projections contain support projections

Let $X$ be a compact space and $\phi$ be a state on $C(X)$ induced by a probability measure $\mu$ on $X$, whose support is the closed subset $S\subseteq X$.

Then $N_\phi$ is the ideal $C_0(X\setminus S)$, which is unital iff $S$ is clopen.

Thus $\phi$ has a support projection iff the support of the associated measure is clopen.

Now, if $X$ is totally disconnected but not discrete, such as the Cantor set, then $C(X)$ is generated (as a C*-algebra) by projections, but since $X$ contains some closed sets which are not clopen, any measure with support in such a set will fail to have a support projection in $C(X)$.