Which island is best for me?

I stayed in old town in Chania and what you want does not seem to be there. I think there are places to rent with pools outside of town which perhaps even have Chania as their address. But it isn’t old town which is apartments and hotels. I don’t think you are going to easily find all that you want in Chania. But perhaps I am wrong. Look how far the places with pools are from old town. Now old town is pedestrianized so not sure where you would park. We didn’t have a car.

Balos is a long ways from Chania and Elafonsi even further. We went to Balos but elected to stay in nearest town rather than day tripping. It is not a place I would recommend with young children. The water itself is very calm but access is a walk down a fairly steep hill. It is a rather harrowing gravel road to get there.

Sounds like Naxos to me.

That is actually what I was thinking too. St George beach is walking distance from Naxos town. It is small hotels that are walking distance to the beach. There are also restaurants right on the beach and we ate at one of them. It was very good. This is where we stayed and we rented a car for a couple days. The beach here is very shallow and there are lots of kids.

There are other places on the coast that have more beach houses with pools. I am not familiar with them but you can look. But I think it was be much easier to drive into Naxos town than Chania .

Yes Naxos is the solution to your wish list. Find a hotel at St George beach. Its about 300 meters long with lots of family run hotels just steps from the beach. Lovely sand beach and very shallow water. Your 5 year old will find new friends on the beach within 5 minutes of arrival.
Naxos town is just a 10, minute strool along the headland. Easy walk into town from St George, Greeks love children even late in the evening the staff will probably be engaging the kids while you enjoy dinner.
Some images of Naxos
Naxos town https://www.flickr.com/photos/stanbr54/albums/72177720299959098 2022

Naxos sunsets https://www.flickr.com/photos/stanbr54/albums/72157700728046481
St George beach https://www.flickr.com/photos/stanbr54/albums/72177720299943096/

to add to stanbr's recommendation, st george is also desirable location because the walk to port town amenities is flat, paved road and sidewalk along shore... so you can use a stroller (bring cheap foldable "cane" type). but do not Dither!! it's already late for booking best locations -- and don't hold out for a pool, you couldn't relax beside one anyway with young kids, no shallow pools.

Naxos it is! Looked into after your replies and it's a dream. Exactly what we were looking for. So glad I asked here and thank you so much for your responses.