I too would like to add scores but I think the concern with the score is that power users could completely dominate the top. e.g. There are users who have been using Anki for more than 10 years and if their Streaks are included in the score calculation they may always come out on top. This problem would not occur in the case of reviews.
There is already an add-on to display Anki’s data in Discord, NameLessGO gave me the info and I forked it (since the author discontinued). It might be possible if I customize this to display the leaderboard data, or embed a function similar to this in the leaderboard.
add-on:
AnkiCord - Discord Rich Presence