Appreciate any feedback on our July 2023 Itinerary

Take a day away from Santorini and add it to Crete. Or drop Crete and take that day from Santorini and add it to Naxos.
Although I love Paros drop the day trip there and focus on Naxos. Choose one or the other, Paros or Naxos, to stay on.

This plan has flaws even if it were a three week trip. While island hopping in Greece logistics are really important for a good experience. You are simply trying to do much in too little time. Every ferry hop uses up at minimum half a day.
Your choice of islands is good but you simply do not have sufficient time to do this trip.

Here is what I would do.
Arriving in Athens by noon you can easily commute to the port and catch the evening blue star ferry to Naxos. Skip Athens on that first day. Keep Athens to the end of the trip. Why commute into Athens twice? Its a total waste of time.

Naxos is a great choice and a great place for kids and to get over jet lag. Spend 4 nights in Naxos to get used to the way Greece works.

Do not bother with a day trip to Paros. Nothing against Paros but why use up a day of your short vacation on a ferry when you have not even begun to see all Naxos has to offer.
Now ferry to Santorini. It is a great destination and a good companion to Naxos. Its all about the views and is worth spending three nights.
Drop Crete it is huge and 2 days there gives you so little time you will not have a chance to have a meaningful experience.

Head back to Athens for the end of your trip. Spent three nights there. It can be done nicely in three days. This plan gives you sufficient time to really explore your islands of choice without being rushed.
Here is why you should spend more time in Naxos.
Around Naxos town 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299959098
Back to Naxos 2022 https://www.flickr.com/photos/stanbr54/albums/72177720299940917
Naxos sunsets https://www.flickr.com/photos/stanbr54/albums/72157700728046481
And Santorini
Santorini https://www.flickr.com/photos/stanbr54/albums/72157632154800782

I agree with Stan. Crete is worth at least 5 nights that you don't have. While the Naxos sunsets were not as dramatic as Santorini's, they were as beautiful as his photos, and more enjoyable on the balcony of my little apartment with a bottle of local wine than jostling with throngs of other tourists or sitting at an overpriced restaurant to see the "show." On Naxos, Agios Georgios beach (St. George) is very kid-friendly. The water is really shallow. I would spend more time on Naxos and plan a couple of day trips to explore the island and less time on Santorini which can be incredibly crowded.

Presently, best I can figure you have 3 nights Athens, 3 nights Naxos including a day trip, 5 nights Santorini , 3 nights Crete and 1 night Athens. If you really want to keep Crete, this is what I would suggest.

Start with Crete and fly into Chania on day of arrival. Stay 5 nights and ferry out of Irakio to Santorini. You can rent a car in Chania and drop off in Irakio.
Stay 3 nights in Santorini. Ferry to Naxos.
Stay 4 nights Naxos, Fly or ferry to Athens (we flew).
Stay 3 nights Athens. Fly home.

This is 15 nights like your original itinerary It saves you a night in Athens at the end. Crete is worth more nights than Santorini. It is much larger and less congested. We just spent 11 nights there and had to make hard choices about what to see so don't expect to see the whole island in 5 days but you can see Chania and visit Knossos as well in that time. We spent 4 nights in Naxos and wished we had more so I am sure you will have no trouble filling your time there. You will appreciate Athens more when you are not jet lagged.

Have to agree with stanbr that due to limited time skip Crete. Also concur to skip Paros day trip. After our RS Greece Tour we extended 10 nights in the Cyclades. We spent 4 nights on Santorini & 6 nights on Naxos. 3 nights would have sufficed on Santorini. If we would have had another 10+ nights we would have considered Crete & Paros. Naxos is a family friendly island with plenty to do. For us it was the best Greece experience. We were there as a couple but enjoyed staying at a resort that catered to families. Enjoy your planning!

I have found the trick with traveling with kids is to allow time. Your children will especially enjoy the swimming in Naxos as the water is shallow and very easy to float/swim in as you look back at the town. You are scheduling only one full day in Naxos. The ferries are fun but you have to allow for packing up and getting to the port, getting your luggage out at your next stop and finding transportation to your new hotel , then checking in and orientating yourself while your children may be tired and hungry. Also while your kids are good walkers it will be hot. Athens is great but I would move some days from the beginning to the end as it splits up some major sightseeing and gets you to the island sooner to get over jet lag and relax and play. Or follow Stan's suggestion and go straight to Naxos although I don't know where you are flying in from

1/ Save your exploration of Athens for the end of the trip.

If it is too late to take a ferry on the day of your arrival or simply to rest one night after a long flight, spend the first night in Athens or in the port of Piraeus and take the ferry the next morning

2/ Remove these 3 nights in Crete.

3/ Remove one night in Santorini (keep 3 nights)
Take a Santorini-Athens flight on 07/24.

By doing this you will have 8 or 9 nights left.

You can keep 5 nights for Naxos and use the remaining nights to go to Paros. A single day trip to Paros is pointless but if you stay 3 nights I can guarantee you won't regret it.
There are several ferries a day between Naxos and Paros, the crossing is very short.
And from Paros you can reach Santorini as easily as from Naxos

Having spent 16 nights in Greece this past May, I agree with the advice from the other posters. We spent 3 nights in Santorini and thought it was perfect. I know some people recommend skipping Santorini because it is so crowded and touristy, but the caldera views are spectacular and well worth seeing at least once in a lifetime. I highly recommend renting a car for a day and touring around the island on your own. There is more to Santorini than just the beautiful caldera views!

I also recommend a minimum of 4 nights on Naxos. And as with Santorini, rent a car for at least one day to explore this beautiful and interesting island.

Regarding the children keeping up and doing walks and hikes, keep in mind the temps will be very hot in July, and this will most likely slow you down. In mid-May, Santorini and Naxos temps were comfortable but by the time we reached Nafplion and Athens, temps were in the 90's. I kept saying to my husband that I wouldn't want to be here during the summer months. I realize you don't have a choice because of your school-age children, but just remember you probably won't be able to sightsee and move around as quickly as you did in Iceland. Make sure you have sunscreen, hats, sunglasses, and plenty of water.

as regards kmkwoo's advice about the kids -- i know that on vacation, days start slow. However if you can, DO try to get the kids underway Early (breakfast by 8, out of apt by 9) for any adventures, sightseeing, really active doings. Then, when the day heats up, you can wind down, with beach time. I've found that works for ALL Ages -- mornings on a ruin or a climb, afternoons in the sea.