Best island for 1 month+ with young kids (2yo, 6yo, 8yo)

Yes Naxos would suit you just fine. Look for accommodation at St George beach. There are quiet neighbourhoods with rental accommodation just back from the beach or perhaps you can find a hotel right on the beach. St george is a good sand beach about a kilometre long in a horeshoe shape. The one end is well serviced with hotels and tavernas and is part of Naxos town. The far end is undeveloped with small sand dunes.

Naxos town waterfront promenade is a 10 minute stroll along the coastal road. You get a beach holiday with all the activities of town close by.
In September everything is still open its warm and the sea is at its best temperature of the year.
Naxos is on regular ferry routes and is also service by air but only in 40 passenger turboprops.

Here is Naxos

Naxos town https://www.flickr.com/photos/stanbr54/albums/72157632094558042
Trip around Naxos https://www.flickr.com/photos/stanbr54/albums/72157634605629689
Naxos sunsets https://www.flickr.com/photos/stanbr54/albums/72157700728046481

Naxos 2019 https://www.flickr.com/photos/stanbr54/albums/72157711294807761

I would also recommend Paros or Antiparos. Both have child-friendly beaches as well as plenty of options for shopping and dining.

I would agree with stanbr that "Naxos would suit you just fine." stanbr gave us the same advice for our Sept. 2017 trip to Naxos! He was spot on. St. George Beach is ideal for a family. We were there for a week and could have spent a month. We didn't want to leave. We stayed at a resort at the far end and it was a 10 min walk to Naxos town. There were mostly families staying at our resort. There is a pool and a restaurant on site. The delicious breakfast was included. Alkyonibeachhotel.gr.

Unfortunately I am not familiar with homes to rent on Naxos.

Try/Google Skiathos island ...
Best beaches in the Mediterranean.
Skopelos and Alonissos islands are very easy to visit.
Great evenings out.

I would also recommend you to check out some of the rest Ionian islands (Zante, Kefallonia, Lefkada) as well as Rhodes island. The important for you is to don t lose time on travelling. These islands have airports and direct flights from Athens

We stayed in an airbnb just off Plaka beach on Naxos. It's an easy bus ride to town, but less crowded and more relaxed than St George. We walked to a different restaurant each night - meaning, your not in the sticks by any means. We have three older kids, and they all enjoyed that beach. Naxos is nice because there is so much to do besides the beach. Driving into the 'mountains' to look at the churches and the marble mine, etc was a nice change of pace.

Thanks all! These are great recommendations and I appreciate it!