Athens, Naxos, Crete, Athens

Putting all the Athens night at the end is more efficient, i.e., save one hotel move in move out cycle.

I would fly onward to whichever island you choose. Having to get from the airport to the ferry port is not what I would want to do after what presumably will be an overnight flight.

Yes, it’s wise to put all your Athens time together at the end of the trip. Give yourself a couple of hours after arriving in the Athens airport, and then fly to Naxos for 4 or 5 nights.

But rather than Crete, I’d suggest going next to Amorgos. It’s much closer to Naxos and a lovely little island. There are two ports on Amorgos — Katapola and Aegali. I stayed in Katapola and loved it but Aegali has better beaches. Read reviews & decide. Spend your 3 nights there, then ferry back to Naxos and ferry or fly back to Athens to complete your vacation.

Save Crete for another trip. It’s a wonderful island — we’re making our 5th visit this year — but it’s very large, further away, and worth more time than you have this trip.

And then save the Athens sightseeing for the end of the trip?

Yes, coming from the USA, it's always a better option to take a flight to an island upon arrival in Athens, provided, of course, that the island has an airport. Save Athens for the end of the trip.

Is there another island better recommended than Crete?

"Better" is a subjective notion; there are more than 150 islands worth visiting in Greece, each one different.

"Not touristy. purposely avoiding the popular islands."

Among these more than 150 islands, Crete and Naxos are in the top 10 most popular and touristy islands. (Crete is by far number 1)

Without knowing why you chose Crete, be aware that Crete is so large that you won't necessarily feel like you're on an island, and 3 nights is only 2 days, far from enough to discover Crete, which requires more than 10 days to be even partially explored.

I would agree with Charlene. Forget Crete, 3 nights is too short. Save those days to add to your stay on Naxos. You could also spend them on Amorgos + 1 night if you can avoid the first night in Athens.
Note, however, that Amorgos doesn't have an airport and it's about a 10-hour ferry ride back to Athens. This is the price you pay for "less touristy" islands. In the Cyclades, an island with an airport is allways very touristy.

Regarding the return trip to Athens, if you want to avoid a 10-hour ferry ride from Amorgos, a viable option would be to return to Naxos (by ferry) and then take a flight to Athens. Of course, this is only because you specifically said "Not touristy, purposely avoiding the popular islands" and you're sticking to that request.

Just in case:

https://amorgos.gr/

Another reason to have Athens at the back of your trip. You don't want to be stuck on an island due to any unforeseen circumstance like bad weather or a strike and therefore miss your flight home.

And another vote to change Crete for another destination. Crete is like a country on its own and 3 nights (which equals only 2 days) is way to short to experience Crete.