Hi,
The system auto selects the first shipping method available based on the the order, and not the first one. So what you're describing won't happen.
However, once a shipping method has been selected, if you add more products to the cart or change the quantity, the system will have to unselect the automatically selected shipping method, and it will reselect another one only if the shipping view is in your checkout workflow.
So it's possible that you would have an unexpected result if you removed it. I however don't know how you setup your checkout or how it looks like with that issue on the website you're doing so I'm just guessing here.