We need at least 10 games for new players to give their rating some time to settle. However, if you already had a rank in the last season you have proven to have enough games, so in subsequent seasons it should be enough to play 3 games to ensure a minimum amount of activity to be shown in the leaderboards.
With 10 games for each rating it would take a lot of time for the leaderboard to fill up again after a new season started.
We need at least 10 games for new players to give their rating some time to settle. However, if you already had a rank in the last season you have proven to have enough games, so in subsequent seasons it should be enough to play 3 games to ensure a minimum amount of activity to be shown in the leaderboards. With 10 games for each rating it would take a lot of time for the leaderboard to fill up again after a new season started.