GUI that should only appear for admins when they join appears for everyone after death?

The problem

You are writing if isAdmin(player) then which means if isAdmin(player) is not equal to nil then it will go ahead.

Solution

You must write if isAdmin(player) == true then which is being precise to what the result is.

local admins = {
    "Player1";
    "Player2";
    "Player3";
    "Player4"
}

local function isAdmin(player)
    for _, v in pairs(admins) do
        if v == player.Name then
            return true
        end
    end

    return false
end

local Players = game:GetService("Players")
local AW = game.StarterGui.AdminWelcome.Gradient

Players.PlayerAdded:Connect(function(player)
    if isAdmin(player) == true then --//This is the change
        AW.Visible = true
    else
        AW.Visible = false
    end
end)

I'm not sure if this helped but I hope it did! Make sure to accept the answer if it worked perfectly. Any problems reply to this answer and I'll try to change it.