Trying to print the LocalPlayer's name doesn't do anything? (it's in a local script btw)

Hi, I'm trying to get the LocalPlayer's name for various things, and as a test I wanted to just print the value of their name. However, with the code I've got, nothing is happening when I run the game.

--ignore the stuff that's commented out below kthxbye

local Players = game:GetService("Players")
local localPlayer = Players.LocalPlayer
--local baseClaim = game:GetService("ReplicatedStorage").baseClaim

function onPlayerAdded()
	local playerName = localPlayer.Name
	
	print(playerName)
	
	--baseClaim:FireServer()
end

Players.PlayerAdded:Connect(onPlayerAdded)

That's the code, it's within a LocalScript that is placed in a normal ScreenGui within StarterGui.

I'm getting no errors or anything from this script, it's just not printing anything.

Thanks for any help.

Client script:

String=Instance.New("String Value")
String.Value = game.Players.LocalPlayer.Name

print(String.Value)

If anything goes wrong let mde know and I'll see to it...