Additionally, dialects were never the language of higher culture anywhere in Italy, all intellectuals have always used a common language and had a nationwide audience even when the country was politically fractured in many states. Said common language being Latin for many centuries after the fall of the Roman empire and Italian from around year 1500.
The above is the hypothetical case of two unschooled people, as said before most if not all italians are bi-lingual with their own dialect and switch automatically to italian (possibly with some local peculiar forms or accent) that is the common language when talking to some "foreigner".
As you say, italians can usually understand (in written form) a dialect because of the common roots, but for what it matters an Italian with some good Latin knowledge can usually get the overall sense of written text in Spanish, Portuguese, Romanian and possibly French.
The unification of language to Italian has been a process that went on for decades, through schooling and later radio and television, when Italy was re-united in the decade 1860-1870 it was a tower of Babel.
Imagine the conversation between a (unschooled) soldier from Friuli attempting to interact with a (as well unschooled) farmer in Sicily in the early years of the unified state.
The differences in language and in culture were definitely reduced only in very recent years AFAIK, and still remain noticeable.
About language, you have to consider how many Italians are (still today) effectively bi-lingual, Italian and local dialect, with the latter ranging from very similar to very different from Italian.