If Frank is Alexander and his plot hinged on Elise choosing a random man on a train full of at least a couple hundred passengers, what would he have done if she'd sat next to someone else?
Probably wait and see if the cops arrest the other man, and then introduce himself in Venice instead of giving her a letter instructing her to bring the "random guy" along to the hotel.
Alexander's plan to ensure that his new identity will never be confused with him relies on the letter to have been incompletely burned and for the inspector to be so bullheaded as to do the arduous work of reconstructing it.
Fridge Brilliance: Frank is horrible at communicating to the Italians, from assuming Spanish and Italian are the same language to saying "Bon Jovi" instead of "bonjourno." This unflattering portrayal of Americans makes sense because Frank is a portrayal of an American by Alexander