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Actually it does not quite, since it is not meromorphic up to the boundary. But it may be an example of the other question, since it has logarithmic singularities which lead to analytic continuation to a complicated Riemann surface.
You don't happen to know if there is a modern account do you? I'm trying to recall a graduate textbook for a "second advanced course in complex analysis" (something like this) that included this topic and lacunary power series as two chapter.
The choice of M is very good. I think a similar statement can be proved if $E_n |X_i|^{2+\delta}$ are bounded with high probability, here you used convergence of $E_n X_i^4$.
