Cinque Terre vs Amalfi Coast

Yes. The CT is an excellent compromise for the AC. Next time you can go south if everyone likes the CT. They are both beautiful and crowded.
The villages are all close and easily reached by train or ferry so you could see them all in a quick day trip.

Given your limited time, the vote was almost unanimous to eliminate the Amalfi Coast from your desired destination list of AC, Florence, a bit of the Tuscan countryside, Rome + Venice over 10 nights. It's really out of the way, and should be saved for when you can give it adequate time.

Really I believe what they want to see are the beautiful pastel
colored buildings lining the cliffs, do a little shopping in a quaint
village and see views of the water.

Yes, it would fit most of this bill. Just be aware that most of the villages + transport system (trains and ferries) are likely to be very crowded so I wouldn't expect a laid-back experience. The golden hours in the CT are the ones before the hordes of daytrippers arrive in the morning and after they leave in the late afternoon/evening. But the AC would be very busy and their transport system stretched to capacity as well.

To have some breathing room during high season, I recommend NOT day-tripping the CT as you'll likely end up in the middle of the hordes unless hiking some of the upper trails. Staying a night or two - which I know you don't have time to do - would allow you to enjoy a less overrun atmosphere. If willing to do some climbing, there are some spots that would likely be less visited - or they were when we were there some years ago - but I won't swear to it.

The CT is also not a place that would be fun on a rainy day.

All that said, people DO day-trip the CT. There is a train from Firenze Campo Di Marte station at 7:54 AM that is direct to La Spezia Centrale in 1 hour and 43 minutes, no changes. From there you'd buy tickets or a day pass for the regionale trains which serve the 5 villages. It's very important to validate that pass or the individual tickets before boarding as checks can be frequent and fines are not cheap. There is a train from La Spezia at 19:48 that will have you back at Firenze Campo Di Marte in 1 hour, 41 minutes. These two trains provide the quickest method in-and-out of the region. If you prefer to depart from/return to more central Firenze SM Novella station, that only adds abt. 20 minutes each direction + 1 change at Firenze Campo Di Marte; also the quickest way via rail in and out of the region via that station.

2024 schedules/prices for the ferries aren't up yet but keep an eye on this site:
https://www.navigazionegolfodeipoeti.it/en/timetables

Which village? They are all very small so I'd definitely do a couple. Vernazza has been reported to be the most crowded these days so look at Manarola, Riomaggiore and/or Monterosso. I was very fond of tiny Corniglia but it involves a very short ride on the small bus or calf-stretching uphill climb via a many-switchbacked stairway to access the village from the train station. No ferry service to that one

Editing to add: I'm sure you were considering the AC as a day trip from Rome, not Florence.

I have not been to CT (only the AC), but I agree with the first reply. Have you looked into the distance and time from Florence? It's much farther than the trip for those travelers who can't leave Italy without seeing the famous ... Leaning Tower of Pisa. To me the destinations don't pair well. But you are right about "northern Italy."

I saw that you are accepting input from kids who don't have to do the transportation, but even if you were starting from Rome, the AC would be an annoying distance for a daytrip. You didn't give the month of the year, which is really important for high-tourism destinations like these. And you did not use the word ... beach? Are they interested because they saw the PBS program about, say, Portofino? What is their concept of what they're going to see on this rushed round-trip?

EDIT: I mentioned the word "beach" because young people may find the attractiveness the chilly Mediterranean to be a subtle thing. The coast of United States, and the Caribbean if your family goes there, have much better beaches (that is, sand and water ...) than Italy.

You also did not mention the word ... Pompeii, which is a very good intellectual reason to go towards the AC, but not for a daytrip from either Rome or (gasp!) Florence. I wish I had seen Pompeii earlier in my life.

You may wish to use this site's Search box top center, because CT is often discussed here, and you might pick up some useful information on climate-damage in the last two years, slow-to-repair trails, and savage summer crowding. I believe that a car is a real burden to have in this area. You'll see you can filter your search results for the last 6, 12, 24 months.

As you can see from Kathy's explanation of the logistics involved, a "quick stop" in the Cinque Terre comes at the cost of at least 4 hours on trains. The payoff will be some time in probably-sardine-like conditions. I just talked with my sister-in-law about her experience in the Cinque Terre 2-1/2 months ago. She had only negative comments about conditions there and specifically felt there wasn't anything interesting to buy in the shops. If you're traveling during the summer, the area will probably be a lot more crowded than what she experienced in October.

You may feel you need to do this, because if the youngsters don't see it, they won't know how disappointing the CT can be these days. It's just that it's a real waste of time when you have so few days available.

Thank you so much to all for your input. I really do appreciate it. I should have mentioned we'll be in Italy 5/16 - 5/26. We'll be arriving via a nightjet train from Vienna early on 5/16, so that is a full day, but we also leave early AM on 5/26 so that is only a travel day.

I'm still weighing our options and trying to decide how to proceed with making everyone happy while also keeping the trip manageable! I'll take all of your input into consideration. Again, it is much appreciated!

Since your group includes young adults rather than minor children, you really could split up for one day and let some folks go to the Cinque Terre while others explore closer to your base (Siena? Lucca? a van tour of Tuscany?). You could get together at dinner and compare experiences.

Given the focus of the rest of your trip, you cannot reasonably include the AC.

CT is manageable from Florence.

so I think it would be quite difficult to go that far south without
wasting a lot of time just sitting on a train.

We stayed in Sorrento for 11 days in April and even from there a lot of each day was eaten up by transportation to see the Amalfi Coast. Transportation in that region is slow, confusing, chaotic, and unreliable. It's still well worth it, IF you have plenty of time. I've never been to the CT, but it makes more sense for this trip.

I should have mentioned we'll be in Italy 5/16 - 5/26. We'll be
arriving via a nightjet train from Vienna early on 5/16, so that is a
full day, but we also leave early AM on 5/26 so that is only a travel
day.

Arriving where in Italy? Even if you are getting into (?) early, you only have 9 FULL days and most/part of another on the 16th. Wherever it is that you are alighting at, you'll also need to deal with the luggage and get your bearings in a new city.

Every location move you make with 6 people will likely eat 1/2 a day. As you can see, even a CT day trip will consumer 4 hours transport time or so. Based on that earlier discussion, what are you thinking your itinerary looks like now? In to where, and flying home from where?

Some possible alternatives:

https://www.travelandleisure.com/trip-ideas/bay-of-poets-ligurian-coast-region-italy

https://girlsgottadrink.com/cinque-terre-alternative-italian-riviera-destinations/

Others can comment on whether better/easier with the balance of your intinerary.