Ha, of course. Thank you. Thanks also for the comments- your suggestion makes perfect sense

Let $(M_{n})$ be a sequence of random splittings corresponding to a random walk on the mapping class group. Then for fixed g, the probability that $M_{n}$ contains an incompressible surface of genus at most $g$ tends to $0$ as $n \rightarrow \infty$. This follows from Maher's "Random Heegaard Splittings" paper mentioned in the post; he shows that the Heegaard distance of $M_{n}$ will grow linearly with overwhelming probability. Then by a theorem of Hartshorn, the minimal genus of an incompressible surface in $M_{n}$ also grows linearly. I think Maher mentions this in the intro of that paper.

thank you for your response- it seems Ramsey's theorem works quite well to answer the finiteness question.

@Misha: I've come across a bit of the literature on carrier graphs, but not in this context. The particular graphs I'm looking at are a union of two simple closed curves. There exists some nice upper bounds on the length of each edge of a minimal carrier graph on a hyperbolic surface (to my knowledge, due to J. Barnard in this paper), but if anything I'd be looking for lower bounds to rule out the possibility that complementary regions can be very small hyperbolic polygons.