Hello Amya! Time to get busy -- island transport AND lodging for fam of 4 in Hi Season already a challenge! But guess what?? I just now spent a few minutes clicking links & I think I may have a perfect answer for you (see last paragraph) -- But u would need to act FAST, to book choice lodging (no-risk, because of free cancellation).
I DO agree strongly that w. kids the age of yours NAXOS is STILL the best choice. IF you can find Lodging. It’s the only island that fills your wish list so well, and is family friendly, tons of teens, they wander at will, safely, & parents can relax.
Of course as others point out, it makes it easier IF you could fly direct from Spain To a Greek Isle — but on other hand that makes the destination that much MORE crowded and believe me, in July the “major-airport islands” will be JAMPACKED Naxos only has a small airport, 2-3 flights a day, and NO cruise ships arriving w. 3000+ passenger (eeek). Thus, while its busy, it's not crammed! And lodging prices will be about half of Santorini.
IF you want to be most time-efficient, you really need to head for island FIRST, immediately on arrival, and do Athens last. And in this regard (remember for the future) it’s important What time you arrive at ATH airport — it greatly affects options. Here’s where you stand w. Flights to Naxos vs. Ferry to Naxos
FLIGHTS - €109 ($125?) p person: small modern plane; on July 4 there still is room for 4 on flights @12:55 and 17:35 (5:35 pm). U need to check in 1 hr ahead. If yr flight from spain arrives late Aft u’d need to stay near airport in PM and take July 5 flight at 6:10 am (not w teens!) or 10:30 am.
FERRY - closest ferry port, RAFINA, about 30 mins by taxi. There are daily fast-ferries (?40€ per?) at 3:45 & 4:45pm (3 hr trip). If your Flight from Spain lands afternoon, you could stay in Rafina for evening (pleasant little town) and get an early AM ferry on July 5. NOTE: if you do decide to use this ferry option, because it's high season, it's a good idea to reserve tickets online thru Amphitrion Agency at airport, and pick up tickets on arrival.
LODGING — Starting this late, for apt in HI Season, it’s going to be almost impossible even w. airbnb, to find an apt for 4 with sep BR, that’s on or walkable to best beaches for you (St George or Ag Prokopios). In Naxos, big places tend to be inland (historical reasons: it started out as v. budget-friendly island, so beachfront filled up w. modest family-run hotels, and when demand rose for villas & apts, they had to be built farther, many not walkable to beach). Thus you'd have to rent a car. Worse than that, kids don't drive. Thus, every time you & husband wanted to relax on patio, kids would be whining to go to beach. Best to have a place that’s WALKABLE to beach. That said, there is a very useful near-beach or on-beach option — side by side “studio units.” This is very nice, esp. for families like yours in a small place w only 8 -12 units. FLASH!! On booking.com just saw there are 2 units left at SUN BEACH HOtel… only a few yards from sand at St. George beach, for July 4 - 9. No advance deposit & FREE cancellation… I strongly urge to book NOW — if you have to “tweak” dates later, no problemo! No-risk situation. (Amazingly, last week I helped another family find Sun Beach for SAME week w 2 kids 11-13!) Even if that's filled up, there are other such units in small lodgings; look at Booking.com St. George beach asking for 2 rooms for 4 adults (yes they count as adults).
PS: Staying on Naxos you STILL could see Santorini's "Famous View" (caldera/cliffside) because Naxos, unlike most islands, has day-excursions to Santorini 2x a week.
PPS: after your island stay, u can take Blue Star Ferry 6 pm to Piraeus (arr 11:30) and taxi to Plaka-area hotel in Athens. U'd have to check out of rooms @noon but hotelier can stash yr bags & it would give u a last afternoon on the beach!