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);
}

Just to add something else I also pass the value back as null e.g.

    public static void RemoveCookie(string cookieName)
    {
        if (HttpContext.Current.Response.Cookies[cookieName] != null)
        {
            HttpContext.Current.Response.Cookies[cookieName].Value = null;
            HttpContext.Current.Response.Cookies[cookieName].Expires = DateTime.Now.AddMonths(-1);
        }
    }

The Cookies collection in the Request and Response objects aren't proxies for the cookies in the browser, they're a set of what cookies the browser sends you and you send back. If you remove a cookie from the request it's entirely server side, and if you have no cookies in the response you're just not going to send any thing back to the client, which won't change the set of cookies in the browser at all.

To delete a cookie, make sure that it is in the response cookie collection, but has an expiration time in the past.