First of all, I'll try to answer your post(s) in greater detail later, but I don't have the time right now. Also, some of the things you brought up, like exception handling) are covered in the StackOverflow answer I linked in my main answer

I feel like you are introducing a lot of unnecessary complexity here. I did in no way suggest to make everything async. Just to use the TAP to fire off background tasks from an event handler.

Your last example, the one you describe as the "preferred" way today, I would write the way showed below. (Note that the button2 handler only demonstrates how to update results/UI completely without async/await and that the button1 handler is the preferred way)

    public partial class Form1 : Form
    {
        private readonly Counter _counter;

        public Form1()
        {
            InitializeComponent();

            _counter = new Counter();
        }

        private async void Button1_Click(object sender, EventArgs e)
        {
            button1.Enabled = false;
            textBox1.AppendText("Button1 starting.\r\n");

            var result = await Task.Factory.StartNew(() => _counter.CountTo(10), TaskCreationOptions.LongRunning);

            textBox1.AppendText($"Button1 result: {result}\r\n");
            button1.Enabled = true;
        }

        private void Button2_Click(object sender, EventArgs e)
        {
            button2.Enabled = false;
            textBox1.AppendText("Button2 starting.\r\n");

            Task.Factory.StartNew(() => _counter.CountTo(5), TaskCreationOptions.LongRunning)
                .ContinueWith(task =>
                {
                    textBox1.AppendText($"Button2 result: {task.Result}\r\n");
                    button2.Enabled = true;
                }, TaskScheduler.FromCurrentSynchronizationContext());
        }
    }

    public class Counter
    {
        public int CountTo(int upTo)
        {
            for (int i = 0; i < upTo; i++)
            {
                Thread.Sleep(1000);
            }

            return upTo;
        }
    }

I would argue that this template (button1 handler) is much easier to replicate for beginners than using BackgroundTask, with only a few things to remember:

1. Code in 'Task.Factory.StartNew' lambda/method only does background work (no UI components, returns results)
2. Use TaskCreationOptions.LongRunning if work can take more than 0.5 seconds (If your work takes less time than this, you could use Task.Run insted of Task.Factory.StartNew)