I don't understand why this simple local script doesn't work with a screen gui?

The issue is that the contents of StarterGui are copied to the PlayerGui, and PlayerGui is what you see. You can't click a button inside StarterGui, it's just there waiting to be copied over. What you want instead is:

script.Parent.MouseButton1Click:Connect(function()
    print("working")
end)

The LocalScript will only run once it has been copied to PlayerGui, and from there it will obtain the button correctly, using script.Parent. Note that you could also do this:

local button = game.Players.LocalPlayer:WaitForChild("PlayerGui"):WaitForChild("ScreenGui").MouseButton

button.MouseButton1Click:Connect(function()
    -- ...
end)

StarterGui just replicates everything to the LocalPlayer.PlayerGui So you should change everything in the PlayerGui and not StarterGui. Put your script under the ScreenGui and not under your TextButton.

script.Parent.MouseButton.MouseButton1Click:Connect(function()
    print("Working")
end)