BetFL - There weren't even 10 other visitors besides us and our guide on the entire site when we visited the ancient Messene in late May (3 years ago). See for yourself
https://photos.app.goo.gl/tTdFDXYjcZSbwTdq6
No need to buy tickets online, just go there directly, there's a ticket booth on site.
I had also considered Ithoumi taverna as a possible stop, but in the end we found other places.
Two tavernas near the ancient Messene, one of which is only a 5-minute drive away in the village of Arsinoi, "To tiganaki tis Xristitsas"
https://maps.app.goo.gl/ZvA6sdQxVeon6Vxt9
We have very fond memories of this one, especially because of the people we met there and because we spent much more time there than planned. So my opinion on the meal is a bit hazy (anyway, we didn't even choose the dishes). After the meal they drag us inside, the after-dinner drinks were on the house, and I don't think we even paid for the meal. LOL
The other one is a little further north, on the E55 road.
Its name is Platanos.
https://maps.app.goo.gl/AG6rvTkjj5EagKhB7
Very nice, traditional taverna, homemade dishes, good grilled food, you eat on a terrace under the trees, about €12 for the meal including house wine.