Don't use string.find to search a table.

table.remove needs where the thing is, not the thing. You need to locate where a value is in the table.

A simple way to search for a thing is to go over each entry in the table using for i, v in pairs(t) and check if v is what you're looking for. If so, then you know that t[i] is v -- so that's where you can remove it from.

function search(tab, val)
	for i, v in pairs(tab) do
		if val == v then
			return i
		end
	end
end

This function returns nil if it doesn't find it. Thus you can do something like this:

local i = search(gamePlayers, leavingPlayer.Name)
if i then
	table.remove(gamePlayers, i)
end

A better solution?

First, I would recommend storing Player objects instead of their names. You are rarely more interested in what you call an object than the object itself (you want its .TeamColor and its .Character and .UserId, etc., don't you?)

Second, instead of constantly fiddling with some list, just write a function that tells you everyone at a particular moment who belongs. For example,

function getPlaying()
	local p = {}
	for _, player in pairs(game.Players:GetPlayers()) do
		if player:FindFirstChild("PlayingValues") then
			table.insert(p, player)
		end
	end
	return p
end

Regardless, playersInGame is not necessary. Just use the length of the list you're building: #gamePlayers