Regarding the islands, both are nice and I wouldnft say they are overly touristic.
This year however there is the triennial art exhibition so there might be more people, but also more art.
The one that is least touristic is Inujima but also full of art.
The Teshima Art Museum is excellent although it has only one exhibit.
Naoshima has more museums and depending how many of those youfll want to see you easily need two full days on Naoshima alone.
For me (slow traveller ) it would be 1 day Teshima, 1 day Inujima and 2 days Naoshima. If you need to commute there I found Takamatsu a good port city which in itself also deserves a day.
The good thing about those 3 islands is that you can sightsee something different from temples and shrines :-)
To Nagoya, I have been 3 (!) times in Nagoya. Once 20 years ago when I saw the castle (nice but mainly rebuilt, there are better ones: Hikone, Himeiji, Bichu Takahashi) , once about 10 years ago where I visited the castle again (with a colleague for whom it was the first time in Japan) and once 2 years ago on a stone day traveling back from Ise to Tokyo when I went to the train museum. http://bicycletraveljapan.blogspot.com/2017/10/day-37-ise-through-nago...
No plans to return, except maybe on a really rainy day that I need to spend between Kansas and Tokyo.
But a lot of plans for other sightseeing in Japan (just went to Miho museum yesterday , which was stunning.... already want to go back in autumn. )
Regarding Osaka I am with the fraction that thinks here is nothing interesting (writing this from my hotel in Osaka where I am now since 3 weeks - business). But everyone is different.
Enjoy your trip to Japan