You don't have to give them access to your emails, that should be clear. They dont just provide this service to Gmail customers !I don't particularly like giving TripIt access to my emails, so I simply created a new Gmail account and linked it to that instead. I can then forward itineraries that I want captured to the new email address.
I just forward the itinerary emails to [email protected] and it works perfectly. What you have done is create an extra level of work for no benefit.