The word that comes to my mind is "crazy".
Have you been to Italy before? Most towns are short on sidewalks and have rather rough streets, with lots of cars driven in interesting
ways.
Wagons are typically pulled behind, which means you have a hard time watching the child and seeing where you are going. Wagons can be harder to control in tight situations. They're longer and bigger than strollers, not so good on narrow, hilly streets or tight doorways. I spent a couple weeks in Italy with my bro, sis-in-law and toddler niece, who lived in Rome. After their move, the first thing they did was toss the American stroller in favor of a locally made "clunker" stroller with bigger, thicker tires and a built-in shopping basket - because that's how the locals took their kids around town.
I'd work on getting the child used to a stroller before the trip.