My proposed train line would essentially be within the airport, so you could check in at either airport and then get on the train to travel to your gate, with baggage traveling in a luggage car.

Alternatively you could ask people to pick up their bags and then get on the train before checking in again. This is exactly what you need to do at Washington and other airports in the US (not so sure about Europe).

New York manages to be a city with two hub airports (poorly connected) and yet it remains a major world city.