Skill-wise Italy is like everywhere in continental Europe, i. e. a couple of notches below the UK level. Yes, the club members tend to be wealthier on average compared to other countries. But I have not found this to be a detriment. Generally, the smaller number of players makes the whole golf experience more relaxed and greenfees aren't any higher than elsewhere. Obviously, there are a few tourist courses (Il Pelagone for example), where it's different - you won't find any locals there, but the usual crowd from other countries.
Personally, I never hesitated to make room for great golf, neither in Rome nor in Florence. Appreciation of the arts and culture does wane a bit, when you're eating dust and moving through crowds for three days in a row. Also, how much can you really remember about a specific sight, when there are about 20 important churches and museums on your itinerary?
My advice is to not overdo the cultural side, pick a few highlights and see if you can get to Ugolino on a bright and sunny day.
Ulrich