The problem here doesn’t really lie with ASP.NET but with the amount of information that is provided in the http request by browsers. The expiry date would be unobtainable regardless of the platform you are using on the server side.
As you have summarised yourself in your question the Expires property of the HttpCookie object that is provided by the HttpRequest object is always set to 1/1/0001 12:00 AM.
This is because this expiry information, as well as the properties such as domain and path, are not passed by the browser to the server when it sends a request. The only cookie information that is sent is the name and value(s). Therefore cookies in the request will have default values for these ‘missing’ fields as they are unknown on the server side.
I would guess the reason behind this is that the expiry, domain and path attributes of a cookie are only intended to be used by the browser when it is making a decision as to whether it should pass a cookie in a request or not and that the server is only interested in the name and value(s).
The work around you have suggested of duplicating the expiry time as another value of the cookie is a way to get the behaviour you are looking for.