You're almost there. You just need to get the TextPattern from the addressEditBox element. Here is a full sample Console app that dumps out all currently running Edge's windows on the desktop:

class Program
{
    static void Main(string[] args)
    {
        AutomationElement main = AutomationElement.FromHandle(GetDesktopWindow());
        foreach(AutomationElement child in main.FindAll(TreeScope.Children, PropertyCondition.TrueCondition))
        {
            AutomationElement window = GetEdgeCommandsWindow(child);
            if (window == null) // not edge
                continue;

            Console.WriteLine("title:" + GetEdgeTitle(child));
            Console.WriteLine("url:" + GetEdgeUrl(window));
            Console.WriteLine();
        }
    }

    public static AutomationElement GetEdgeCommandsWindow(AutomationElement edgeWindow)
    {
        return edgeWindow.FindFirst(TreeScope.Children, new AndCondition(
            new PropertyCondition(AutomationElement.ControlTypeProperty, ControlType.Window),
            new PropertyCondition(AutomationElement.NameProperty, "Microsoft Edge")));
    }

    public static string GetEdgeUrl(AutomationElement edgeCommandsWindow)
    {
        var adressEditBox = edgeCommandsWindow.FindFirst(TreeScope.Children,
            new PropertyCondition(AutomationElement.AutomationIdProperty, "addressEditBox"));

        return ((TextPattern)adressEditBox.GetCurrentPattern(TextPattern.Pattern)).DocumentRange.GetText(int.MaxValue);
    }

    public static string GetEdgeTitle(AutomationElement edgeWindow)
    {
        var adressEditBox = edgeWindow.FindFirst(TreeScope.Children,
            new PropertyCondition(AutomationElement.AutomationIdProperty, "TitleBar"));

        return adressEditBox.Current.Name;
    }

    [DllImport("user32")]
    public static extern IntPtr GetDesktopWindow();
}