  We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational
Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an
assumption, that 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
conditionally bound degree $d$ points not pulled back from points of degree
$d/2$ on the projective line. Furthermore, we show one may take $B_2=24$ and
B_3=114$.
  Our proofs proceed by refining recent work of Park, which applied tropical
geometry to symmetric power Chabauty, and then applying results of Bhargava and
Gross on average ranks of Jacobians of hyperelliptic curves.
