No, you don't have to avoid Rome — it's not as bad as Venice, and can support more people — but plan ahead and don't just do a tour of all the 'must see' highlights. Look into the off season if you are a history buff with a hyperfocus on Rome — you won't be able to finish your list otherwise due to all the pointless waiting around.

And yes, visit provincial villages and eat in an authentic Italian restaurant where tourists are mostly other Italians. Experience the difference. But you are not limited to villages. Italy is huge, and there are a lot of cities with remarkable museums, world-renowned festivals, great cuisine, and where your money is more than welcome and your stay won't be marred by extreme crowds and pushy con artists in faux Roman gladiator gear.