I didn't dare answer earlier as the OP was asking for an island with resorts and most small non-touristy islands have none (they all have plenty of restaurants, though ). At any rate, there are lots of great non-touristy islands in the Aegean (and Ionian), Symi and Kastelorizo being two of them. Others that come to mind are Thasos, Samothraki, Skiathos, Alonissos, Skopelos, most Cyclades except for Mykonos, Ios and Santorini, which are extremely touristy, and most small members of the Dodecanese, except for Kos and Rhodes for the same reason as above. Also, all of the Ionian outside July and August and not Corfu, which is also packed from April till October (some resorts on Zakynthos and Cephalonia).

A word of caution: Aegean is a member of *A but Olympic that flies from ATH and SKG to the islands is not. You do get M+B miles but for any other *A programme!