I would like to make sure the following proof is correct.
Spectral theorem, simplest form I will be using-
Let $H$ be a separable Hilbert space and $T:H \to H$ normal, then we may wlog assume $H = L^2 (X,u)$ where $X$ is a second countable locally compact Hausdorff (these properties are easy since $X$ will be a countable union of compact subsets of $C$), and $u$ is a complete probability regular measure (the Riesz-Markov gives that it's complete so let's take that for fun), and $T$ acts by $M_f$.
Von Neumann's ergodic theorem-
Suppose $T : L^2 (Y,v) \to L^2 (Y,v)$ is unitary, let $U$ denote the closed fixed subspace of $T$, and $P_U$ the projection. Then for any $h \in L^2(Y,v)$,
$ \frac{1}{n} \sum_{i=1}^n T^i(h) \to P_U(h)$.
Proof-
Since unitary is normal, we use the spectral theorem to transfer our world to that of $H=L^2(X,u)$, $T = M_f$.
Since $T$ is unitary, the spectrum lies in $S^1$ (this is because it is contained in the disk of radius $1$, and $T^*(T-r) = 1-rT^*$, and the right is invertible when $|r|<1$ by the geometric series). This lets us assume wlog that $f$ takes only values on $S^1$ (because where it doesn't, is of measure $0$, since the support of the pushforward of the measure to $\mathbb{C}$ is the spectrum).
Now, $U$ is identified as the functions with a representative supported in $f=1$, and multiplying by the indicator of $f=1$ is thus the projection.
Now for any fixed function $g \in L^2 (X,u)$ $\frac{1}{n} \sum_{i=1}^n f^ig \to P_U(g)$ pointwise, by DCT (dominated by max($10g^2,10)$) this gives the desired convergence.
As long as the Spectral Theorem ensures that $X$ has finite measure, your argument looks good to me.