So if it was actually with the handprint card, I wonder if it's different now with the passport?

I have a feeling you're not the first person in the world to ask that question. Why not give it a shot and let us know how it works out? ;-)

Doesn't control of the ticket to over to the airport from the travel agent?

I recently learned that lesson too. I was about to board a flight that got delayed for three hours and at one point it seemed like they are going to cancel the flight. Now this was an early morning AZ flight and there were three later flights with seats available. The agent at the gate told me that in case that the flight is cancelled they will reroute us but in order to do that everyone have to go land-side and start everything from the beginning. That would have been a nightmare. He said these are the procedures in TLV. At the end I boarded the flight although I already knew I'm going to miss my onward (unprotected) connection but this is a different story :)