Tonsil stones can easily be removed on your own - either you use a long object like a chopstick and apply pressure, or you wiggle around your tongue. Gargle with mouthwash first to loosen them and it should come out easily.

As far as the specialist's recommendation goes, removing your tonsils will prevent future irritation from tonsil stones. However, the tonsils exist for a reason - they catch dirt and prevent further infections such as sore throat from occurring. The worst tonsil stones do is give you bad breath, and you can work around this by just practicing proper oral care. You don't need to have them removed unless you regularly get tonsillitis.