In the past there was a very useful day excursion on the vessel MV Alexander -- I believe it was on a Tuesday or Wednesday. It left Naxos Harbor at a little before 8:30 am , stopped at Paros to pick up passengers, and arrived at Santoirni port at 10:30. THere the passengers are met by a bus with a licensed guide (you may follow tha guide at the destinations, but are not obliged to at all). First the bus takes the group up the steep road to the top... the living area of the island, going straight thru the main town (Fira) without stopping, and immediately to the Village of OIA at the northern tip -- the place that's seen on all the calendars, postcards etc. The group has about 2 hours there to wander, take photos, buy stuff, sit in cafes, etc. Then, if you return to the bus, you are taken back to Fira at about 1 pm, with free time for the remainder of your stay. You can visit the Museum, walk up & down the rimside path for the view, as you did in OIA, and find a place for lunch. The excursion boat leaves about 4:30, so you must return to the bus a little before 4 pm to be taken down to the port. The boat will get you back to Naxos harbor about ? 7pm. There is food & drink aboard. The cost used to be about €54 per person.
If you want a day, just to see the Famous Caldera View, but do not wish to stay overnight, this is the handiest alternative -- IF IF IF it will run in summer of 2023. None of us are sure how the Covid never-ending pandemic will affect things.
LOgistically, the very simplest & most time-efficient way to get a glimpse fo Santorini would be this: 2 hrs after you land in Athens, get on a domestic flight to Santorini ... and find rooms that are near the cliff edge. They don't have to have a view from your balconies (unless you have deep pockets)... just be a short walk from the rimside path. Enjoy the famous Caldera View at sunset (NOT in Oia because the rim-path is a ZOO there), the next AM, go early to OIA 8 am, before the cruise-ship and package-tour crowds arrive .... then take the 3:15 pm ferry to Naxos. Save your swims for Naxos... that's where the beaches are.
As to your original plan to ferry from Athens to Naxos --- we cannot advise you on whether to do that, when we have no idea about the particulars of your trip. Dates of trip, length of stay etc ., We can't help with generalaities like "summer of 2023" If you have 3 weeks to spend, maybe it's just fine with you to spend half a day on a ferry .... the best ferry leaves at 7:30 am arrives about 12:30. Incidentally, as a parent myself back in the day ---- I don't know how good you are with enforcing military-style obedience on teens -- but when mine were that age, I can't imagine being able to get them out of bed, dressed, packed, and out of a hotel in time to catch a 6:30 am Metro to Pireaus Port, to board the Blue Star.
If you have only 10 - 14 days total in Greece, I would not do Athens at the beginning and at the end of your trip (you need to be there at the end). I would head for your island (Naxos or Santorini) immediately upon arrival in ATH airport --- if your flight lands by mid-morning, you can get a domestic flight at 12:30 or 1 pm, and... if going first to Naxos, you can be jumping into the sea by 3 pm. If your flight lands early afternoon, and you still want a 5-6 hour ferry, you could take a bus from Airport to Pireaus to get the 5:30 Blue Star, whichj will arrive 10:30 pm. Again, if I had 4 teens, I would not relish that loooong trip at the end of a looong flight.
You have a lot of research to do, to achieve a trip that will keep everyone happy, and still give parents a chance to relax. I'm glad you are thinking so far in advance, and wish you the very very best of luck.