If you want to do some empirical study, there is a version of KataGo modified for Hex, and KataGo allows training networks that work for multiple board sizes, but I don't know if the code uses the number of moves as an auxiliary target in addition to winning. Networks trained on up to 27x27 board are available here (together with GUI) and the code is here, but apparently the author has started working on an update.
Curiously I arrived at this question by searching "percolation theory" and "Hex". I saw the empirical study of the winrates of first moves by the same author on Zhihu, and wonder whether there's a percolation theoretic explanation of the observed pattern, and whether one can exhibit a winning first move on every (large enough?) board and prove it's winning; because there's always a winner, one only needs to show it's "at least as good" as all other moves.