Cookie is not deleted

Clearing the cookies of the response doesn’t instruct the browser to clear the cookie, it merely does not send the cookie back to the browser. To instruct the browser to clear the cookie you need to tell it the cookie has expired, e.g.

public static void Clear(string key)
{
    var httpContext = new HttpContextWrapper(HttpContext.Current);
    _response = httpContext.Response;

    HttpCookie cookie = new HttpCookie(key) 
        { 
            Expires = DateTime.Now.AddDays(-1) // or any other time in the past
        };
    _response.Cookies.Set(cookie);
}