I just returned from a cruise on the Seabourn Odyssey which visited Patmos. It was my second time there. Other than visiting the Monastery where St. John is said to have written the Book of Revelations there is not much there. Having visited once before, on the Seabourn Spirit, and "done that" along with a walk through the village next to it, this time we just walked through the small town, stopped at a nice wine shop and then enjoyed the ship. Seabourn's tour works just fine and there is nothing special that a private tour can offer.
Mykonos is really dependent on what you want to do, but tours really are not necessary. If you go to the beach, grab a taxi. If you want to wander the town or sit in a taverna in the port and people watch, you don't even need that. (Credit cards are recommended, as Mykonos is expensive.)
Rhodes is a nice place to wander through the walled city and have a nice lunch. (I know a very nice restaurant hidden in there.) Or you can hit the beach in Lindos...just a taxi ride away.
Mylos is quiet, but it is usually a Marina Day for Seabourn so enjoy the toys!
Kusadasi is not an island. What Seabourn will treat you to is Ephesus at night - a private event with chamber music and cocktails. Very nice. During the day, a visit to Ephesus is a must. The city is fascinating. You can hire private guides (I know some good ones) or you can do it through the ship. Either way it will be crowded...but worth it. Want something different? Visit Sirince (pronounced Sur-in-gee) which is about 45 minutes away by taxi. No guide is needed.
Dikili is really best done with the Seabourn tour as a private tour will be pretty much the same thing.
Hope this helps!