I can't comment on Italy, but
Day 7: Fly to Athens - Athens, Night 1
Day 8: Acropolis, Acropolis museum - Athens, Night 2
Day 9: Day trip to Delphi by bus - Athens, Night 3
Day 10: travel to Santorini - Santorini, Night 1
Day 11: views, ending our trip relaxing, walking through Oia - Santorini, Night 2
Day 12: ferry to volcano - Santorini, Night 3
Day 13: Travel home

Some food for thought...
"Travel home"is flight from Athens or Santorini or what? If Athens, what are you going to do in case of a ferry cancelation/delay due to bad weather/machanical problem strike?
Are you aware that Athens city center to Piraeus is half hour on metro,then ferry to Santorini 6 to 8 hours? Then about 20 minutes bus from Santorini port to Fira town,other bus to Oia 30 minutes...
Athens airport to city center about 45-60 minutes on bus. A flight from Athens to Santorini 30 minutes or less, but you have to concider time to commute from/to airports, waiting time etc...
Athens city center to Bus station B 15-20 minutes taxi ride on early morning, 3 hours bus ride, if dropped on Delphi bus station 15 minutes back to museum and sites...
Itenerary is very very fast paced, risky and not making justice on any of your destinations.
At least check Aegean or Easyjet or similar and fly Rome to Santorini. Even going throu Athens. Save Athens for the tail end of your trip so you do not risk your outbound flight so much. If you do not fly out of Athens, arrange to be near your whatever exit point of the country,at least one night in advance.

We are flying round trip Athens to Santorini, getting back to Athens around 8pm the night before our flight to the US. The mainland part of Greece will probably feel a little rushed, but I think we’ll have plenty of time in Santorini. That’s 2 full days, plus the better part of the afternoon our first day and a good half day before flying back to Athens.

Why not explore the lovely area around Rome instead of going all the way down to the Amalfi Coast? We loved our trip in that area but were there 2 full weeks. In my experience, boats and buses are far from reliable and you'll be spending a LOT of time in them and trafic jams are frequent.

Why not stick to Rome and lovely Florence (1.5 to 3 hours by train instead of about 4 hours++) for the Italian part of the journey?

I agree that Delphi is too far for a day trip. I would propose to take a boat from Athens to the lovely island of Poros (1 hour by boat if I rememer correctly!), then rent a car for the day on the mainland and visit beautiful Epidavros and charming Nafplio. It was our favorite part of our last Greek trip...

And Capri, for what it's worth, was a letdown for us and we were there 3 days. For a day-trip, you don't get to see much! Throngs of day-trippers, and long hours in buses around the island... We much prefered Amalfi and Ischia.

It depends on what you want to see/do on Santorini and on what pace. Most of caldera/volcano excursions start at morning or near noon and end up either with a sunset dinner, either early afternoon return, after stopping on hot springs/visiting volcano and stopping for a couple of hours hiking, usually going to Thirasia island for lunch... So on volcano day, you can do not much more than expore a bit of Oia (assuming this is where you'll be based).
1 full day plus maybe 2 half days can be spent (?) exploring Fira town with its small but interesting museums, hiking up the rock to ancient Thira above Kamari, visiting flat side of the island (Perissa,Kamari...), visiting a winery, going to Akrotiri and Red Beach, join a boat tour to white and red beach,visit Mesa Gonia,Exo Gonia,Pyrgos, hike the Oia-Fira path,relax by a tavern taking in the views drinking the local wine and eating some of the local specialities and so on....or not! You definately have no time to see and do it all, you have to prioritise..And remember that bus hub on Santorini is Fira. If you are located elsewhere you have to commute there to connect elsewhere. Otherwise it might be handy to hire a car.