Output saying that argument is missing or nil?

The issue is with the FireClient. FireClient's first parameter is the playerInstance to fire it for. If you want to fire it for the player who joined, make it:

-- server script
game.Players.PlayerAdded:Connect(function(player)
    game.ReplicatedStorage:WaitForChild("LeaderboardEvents"):WaitForChild("RemoteEvent"):FireClient(player) 
end)

Then, you can pass any other stuff through that you need after player. When listening to it on the client, the first parameter received is not the player. So for example:

-- server script
game.Players.PlayerAdded:Connect(function(player)
    game.ReplicatedStorage:WaitForChild("LeaderboardEvents"):WaitForChild("RemoteEvent"):FireClient(player, "StringParam1") 
end)

--localscript
local players = {}
game.ReplicatedStorage.LeaderboardEvents.RemoteEvent.OnClientEvent:Connect(function(param1)
    print(param1); -- "StringParam1"
    table.insert(players,1,player.Name)
    script.Parent.Text = players[1] or ""
end)

This is because LocalPlayer can be used in LocalScripts, but not on ServerScripts. Learn more here.

EDIT: Added developer docs link.