Why wont the textbox's text be the name of the team?

You've got the right idea but if anything you'd have to FireClient (as well as adding it to the StarterGui) to add it to the PlayerGui too.

-- Client
local Button = script.Parent -- obviously relocate this to your button
local Players = gameGetService('Players')
local Player = Players.Player
local replicatedStorage = game:GetService('ReplicatedStorage')
local teamEvent = replicatedStorage.Events.TeamAddedEvent

Button.MouseButton1Click:Connect(function()
    -- you can use this dicationary to add more info on the team later on if you want
    local teamInfo = {
        teamName = teamName.Text -- obviously relocate to your textbox
    }
    teamEvent:FireServer(teamInfo)
end)

teamEvent.OnClientEvent:Connect(function(Team)
    -- with this you can update the UI for all players, keep in mind you sent over the actual team so all properties about the class can be seen here too, color, name, etc
    local newTeamButton = Instance.new('TextButton')
    newTeamButton.Parent = Player.PlayerGui.UI -- change this to your ui parent obviously
    -- more properties here :)
end)

Then from here, connect to the Server.

-- Server
local teamService = game:GetService('TeamService')
local replicatedStorage = game:GetService('ReplicatedStorage')
local teamEvent = replicatedStorage.Events.TeamAddedEvent

teamEvent.OnServerEvent:Connect(function(Player, teamInfo)
    local newTeam = Instance.new('Team')
    newTeam.Parent = teamService
    newTeam.Name = teamInfo.teamName
    -- you should add a check here for a random team color generator for a team color that isn't taken
    newTeam.TeamColor = BrickColor.new("Bright bluish green")
    teamEvent:FireAllClients(newTeam) -- fire back to all the clients
end) -- OnServerEvent

use text:GetPropertyChangedSignal("Text") instead of text.Text