There's no static playbook here, you're trying to play society like an instrument.
Figure out why a player would be motivated to share and then make that action as obvious and easy as possible. For example players share Wordle results as if to start a conversation with ppl they already engage with: "I bet you can't beat my performance". Or, "look at my proud performance." Wordle optimizes this impulse in two ways: it fits the request into a tweet, and it uses a novel presentation to compete with all the other content posted to Twitter.
Novelty is a huge component in that last bit. Once twenty games are spamming Twitter with emoji posts, they'll all suffer. Think Facebook in 2011 when Zynga had everyone's feed stuffed with nails and wood boards and goats.
I would start thinking about the intrinsic rewards your game offers. If it makes players feel smart, they may be tempted to share to humblebrag. That's half the equation (and why focusing on sharing isn't enough) because now you need what they shared to be as enticing a click as the other content on the platform.
Makes sense to me why leaderboards didn't work. Nobody cares to compete with total strangers and even friend leaderboards are extrinsic motivation. It will only speak to competitive player types. Plus if you have any level of success, hackers will ruin your board by submitting unreachable scores.