Can you please help us choose the right island?

We liked Naxos which checks off a bunch of your wants. It was recommended by one our Rick Steves guides as his favorite. You might look at that. Of course we were there in April and it was very quiet so I don’t know what’s the crowds are like in September.

Amorgos , the island from the movie "The Big Blue", ticks almost all the boxes (except that it's in the Cyclades)

Beautiful/ unspoiled hidden beaches/coves to swim and snorkel in

One of the best diving clubs in the Cyclades Islands : https://www.amorgos-diving.com/

as opposed to a beach with hundreds of umbrellas lined up

It doesn't exist in Amorgos.

Gorgeous natural scenery

It's full of them

'Nightlife' being some cute restaurants perhaps with local music and
or dancing

There are also some

no crazy party islands

No chance of finding this in Amorgos

Any old churches/monasteries/ruins would be a big bonus

Like this one ? https://maps.app.goo.gl/Tn1vAY51u5stgL4M6

More information:

https://amorgos.gr/

A second vote for Amorgos. There are two ports on the island — Katapola & Aegiali. We chose Katapola, stayed in the wonderful Porto Katapola hotel right on the harbor. No crowds. Great food. You can do a lot of walking/hiking, maybe rent a car for a day or two to explore the island.

Amorgos does look amazing- but only because we were in Cyclades two years ago, we were hoping to explore a different chain of islands this time around…

I’d suggest Andros. It’s also in the Cyclades, which might rule it out, but it’s the most northerly of the group and quite distinctive.

It ticks most of your boxes: quiet with very few tourists and no mass development, good restaurants and beautiful nature.

There is a scuba club in Gavrion which runs boat trips.

https://scuba-andros.gr/boat-trips-in-andros-island/

The best place to stay though is Chora, the mainly pedestrianised town capital. It is on a headland with beaches on each side, one completely undeveloped. From there you can hire boats to visit various beaches. (Full disclosure: in Andros we always stay in rooms rented by Yanni and Caroline who run the boat company.)

https://androslocation.com/boats-in-andros/

More on Andros and Chora

https://www.greeka.com/cyclades/andros/

https://www.greeka.com/cyclades/andros/villages/chora/

Another advantage, for us anyway, is that the Andros boat goes from Rafina not Piraeus. Rafina is about half an hour from the airport in the opposite direction to Athens. The boat is then two hours to Andros and there is an hour scenic bus ride across the island to Chora.

Depending on timings you may need to overnight in Rafina, but that’s no hardship. It’s a nice little town with good restaurants round the square and a nice hotel - the Avra - which runs a reliable free transfer to the airport.

https://www.hotelavra.gr/

Take a look at Zakynthos, Lefkada and Kefalonia (beaches). I’ve been told by a representative of the tourist board of Kefalonia that it doesn’t get too busy with tourists on the island of Kefalonia simply since they haven’t got that many accommodations to be overcrowded. Never been there but people I know that have been there are very enthusiastic. I like the less touristy places. Kefalonia and the little island of Ithaca are on my bucketlist 😊.

I would also recommend Amorgos as it's a wonderful Greek island and one of my top 3 favorites.

Check out my photos:
https://tinyurl.com/y7bqwfma

Tinos is another gem with some of the most beautiful villages I've been to in Greece. In my top 3:

Check out my photos:
https://tinyurl.com/y6w5ncb8

The other top 3 favorites is Chios. Might be the last large Greek Island not over run with tourists and fantastic scenery including the Mastic Villages:

Check out my photos:
https://tinyurl.com/ycgkrqza

There are numerous Greek Islands that don't get the hype of the Big Name Islands but you'll need to do your research to find one(s) that meet your needs and don't look at glossy magazines or rely on travel agents to tell you where to go!

Every single one.

Go with whatever is easiest to get to.

I mean, I could tell you the 12 I've been to and I would be recommending them because they are the ones I went to. Each and every one of them would qualify under the above criteria.

I’d suggest Sifnos even though there are busier areas than others and they do get the occasional celebrity visit, such as Tom Hanks last June, 2 days after we visited. The food is amazing and people go there just for that. The beaches are amazing, over 100 kilometers of hiking trails. We stayed in Faros, up a steep hill with views of Chrisopiggi. We fell in love with it instantly, but this year we’ve been looking at Ikaria, which is very different from Sifnos. What’s even funnier is that we live on an island we’re constantly vacationing on islands. Whatever you choose, good luck and enjoy!

I spent 2 nights and most of 3 days on Crete. 6 to 7 nights would be a good amount of time for Crete. Santorini and Rhodes look appealing. I haven't been to either yet. You probably want Rhodes because you listed it but you didn't list Crete. Start by seeing how many days Rhodes will need. Crete seemed like the best island for museums, ruins, sights, and so on.

I agree with suggestion of Crete. We spent 11 nights and could have spent far more.

The southwest coast is very underdeveloped and very special. You can only get to parts of it by ferry.

Chania is like Venice without canals. Very charming.

We found the food to be almost uniformly good. And well priced.

We enjoyed Rhodes. There are Rhodes Town and Lindos with many places to stay and dine.