the slideout-exit sits inside the .slideout-overlay – you would need to bring that forward with z-index. Problem is it will then cover the navigation. You could make the overlay transparent and add a pointer-events: none; property to it so you can still click the menu.
The Menu Text can be changed in the Customiser > Layout > Primary Navigation.
The JS is running, i think the problem is because you have added a button to the label, so the click is being registered on the button. You can try removing the slideout-exit class from the areas you have it and and add it to the button.