Hi, Travelmoments, and welcome to the forums!
You've gotten lots of great advice, although some of it may differ depending on personal preference. There are no wrong answers unless it's wrong for you and your tribe. Speaking of that tribe, I'm guessing there are 4 of you?
I'll go along with dropping the CT but keeping the same number of total trip nights; just add them to something else. I'd also do Siena as a day trip from Florence, and you sure could add even another night or two to that one for additional day trips. For instance, the boys might enjoy riding bikes around the old walls of Lucca (1 hr, 20 minutes by train from Florence). Anyway, cutting 6 pack-up-and-move days down to 4 might be wise, what with 4 people to move. It can be wearing and time-consuming to make sure everything got back in the cases, get checked out, to the train, find your next accommodation (in a strange city), get checked in and settled....All before even setting out to do what you came for!
Those longer stays in one place also make apartment rentals (versus hotel rooms) more economical, with the cleaning fees spread over 4 or 5 days. Again, with 4 of you, the extra room would likely be welcome, and you can save some $ with a kitchen for having breakfasts in, maybe some lunches and take-out dinners too. Most teenage boys I've known have been sort of bottomless pits so keeping them sufficiently fed and watered is no mean feat! HA! You'll also need a few hours here and there to get the laundry done: even if you do end up with a washer in a rental apartment, I'll recommend going to a coin laundry as you can get more of it done faster in the bigger machines. Apartments might also have a washer but not a dryer.
Crowds? Grin and bear 'em. If you want to see the Main Attractions, you won't be alone. Yes, get advance tickets for the biggies (we can help with the websites for doing that) and try to do the outdoor ones early in the morning to help manage the heat. Museums sometimes are less busy in the later afternoons. Thing is, busy or not, if you are interested in the treasures a city/region has to offer, it's well worth it to go. All of them will also have quieter corners that fewer rather than more tourists explore: a very good reason for adding more versus less time in the cities: mitigate crowd exhaustion with a daily balance of more versus less-visited sites.
Amalfi Coast: with teenage boys, I'll recommend Sorrento. There isn't enough for them to do in, say, Positano ( I was bored with it in 1/2 a day). There has also been some love for Salerno so that's a possibility too? It's just a bit far by ferry from Capri, if interested in that one, and there's some nice hiking on the island that I think your young people would enjoy. Anyway, Sorrento (technically not the Amalfi but is right next door) is a good transport hub for the region, with bus, commuter train and ferry services. Salerno has all three has well, including high-speed train services from other Italian cities.
Rome? I love the Eternal City. Most folks I know who didn't like her only planned for a couple of days and she's too old and complex a lady to begin to get to know in that short a time. She's sort of like an iceberg, with immense amount below surface. Your crew might enjoy a long walk on Appia Antica? That's a terrific piece of history, and encompasses everything from ancient catacombs to the ruin of an Emperor's circus to the crumbling remains to once-grand tombs: it served as cemetery as well as a military and trade route.
https://www.parcoappiaantica.it
You get the idea... Lots to see and do. You're going to have a GREAT time!