Please make sure you format your code properly, because it is super hard to read in the current state. Here’s a good explanation how to do that:
What I managed to notice is that you are emitting your start_game signal, but you don’t connect to it anywhere. That’s why clicking the button doesn’t do anything. Also make sure to connect the button’s pressed signal to the _on_start_button_pressed method, as I think you haven’t done that either.