We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d<g$ is odd, we prove there exists $B_d$ such that a positive proportion of these curves have at most $B_d$ points of degree $d$. If $d<g$ is even, we similarly bound degree $d$ points not pulled back from points of degree $d/2$ on the projective line. Furthermore, one may take $B_2 = 24$ and $B_3 = 114$.
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Our proofs proceed by refining recent work of Park, which applied tropical geometry methods to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves. This is joint work with Joseph Gunther.