My Wife and I had a slightly different experience at FCO. Last month we flew on miles from ATL-MAD, then connected on Alitalia to FCO. Reason: fewer miles for D1 with the connection. Added a few hours, but experience was much better. First was the fact that our bags were checked through to FCO. Passport control at MAD was similar to what the OP said about FCO, barely a glance and then sent on our way. Nice lounge in MAD where we waited a couple of hours for the connection. The AZ flight was considered "internal" (within the Schengen area), so when we arrived in FCO Terminal 1, it was like landing in RDU from ATL. Walked to baggage claim where no extra scrutiny was made, no customs declarations needed, and then out into the general area. The trip back was a little more hassle, as it required an overnight in MAD (again due to the reduced number of miles). But the interesting part of that trip was the Schengen AZ lounge at FCO had an A320 simulator running in one end of the room, with an actual pilot there to let guests "fly" the plane. Wow, was that great!