Why do callbacks of ContextActionService:BindAction stop working after death?

I created a simple ShiftToRun LocalScript (pasted below).

It works fine, but after death + respawn it stops working, if I put it in: StarterPlayerScripts, or ReplicatedFirst.

However, it keeps working after death, if I put it in StarterGui.

I have a few questions (if I may?):

1. I could not find any concrete info on this, but it appears that the LocalScript is destroyed (and thus the callback is removed from ContextActionService) on death for some reason. Is that about right?
2. More importantly, is there a clear guideline on how to persist LocalScripts after death? Is StarterGui the only way to persist for a script after death?
3. And, in general, when to put what kind of script where? What are the pros and cons of all the different locations? Is there any general guideline or a tabular overview somewhere?
4. What about general overview over script lifecycles? When and how long do they live? What happens when they get destroyed?

local player = game.Players.LocalPlayer
local cas = game:GetService("ContextActionService")

local runSpeed = script.RunSpeed.Value

local isRunning = false
local char = player.Character or player.CharacterAdded:wait()
local humanoid = char:WaitForChild('Humanoid')
local walkSpeed = humanoid.WalkSpeed

local toggleRun = function(actionName, actionInputState, actionInputObject)
	--if actionInputState == Enum.UserInputState.Begin then
	isRunning = not isRunning
	if isRunning then
		humanoid.WalkSpeed = runSpeed
	else
		humanoid.WalkSpeed = walkSpeed
	end
	--end
end
 
cas:BindAction("toggleRun", toggleRun, true, Enum.KeyCode.LeftShift)