Tools in StarterPack are simply there to be (automatically) cloned into each player's backpack.
Also, you are inconsistent with where the leaderstats folder is stored.
In the first section, you parent the folder to the player. In the second section, you are getting the leaderstats folder from game (you were right in the first section, it should be parented to the player).
The best way to add points to the player's leaderstats would be to use a RemoteEvent.
To do this, you would fire an event from the client (local script) to the server and request points to be added to the player's leaderboard stat.
Server-side:
local pointsEvent =
event.OnServerEvent:Connect(function(player)
player.leaderstats.points.Value += 1
end)
Client-side:
local tool = script.Parent
local pointsEvent =
tool.Activated:Connect(function()
event:FireServer()
end)