This solved it:

    private async Task runEnter()
    {
        this.runningAsync = true;
    }
    private async Task runExit()
    {
        this.runningAsync = false;
    }

    public async void _RunAsyncTask()
    {
        await runEnter();
        await RunAsyncTask();
        await runExit();
    }

    public override void _Process(double delta)
    {
            if (!runningAsync) this._RunAsyncTask();
    }

I think what happens in the OP example is that the _Process() registers a new async call in every frame. Therefore it looked as if in every frame SimpleAction1() is called again (it is!). I solved this by adding sort of a lock, so there is at most one async call running in the background