Definitely lots of ruins, churches, and a seemingly infinite number of Madonna and baby Jesus paintings! I am so over art museums!
Since we do private tours, though, we can pick what we want to do, especially with the kids, which is often off the beaten path. We did a wonderful street market food tour in Sicily. We did a trip to hot springs in Valencia. Sardinia was lunch at a local goat farm then swimming at a pocket beach. Our first trip to Rome was a whirlwind check-all-the-boxes, but our second, on the cruise, we skipped Rome entirely and went to Civita Bagnoregio (sp?) It's also a old ruin/city, but accessed only over a really long bridge, and had a nice coffee and pastries in their piazza.
In SPB, we went to a cat rescue home. Had Viking family portraits in Reykjavik. And you get the idea. There are lots of wonderful, slightly off the beaten path trips you can take that gives you a good idea of the culture, but do not include ruins, churches, or art museums.
And yes, most people do spend more time than just the cruise once they get to Europe. In fact, I think the tendency is to try to do too much, since you're already there.
But, IMHO, flying to Europe, to spend
DCL prices for a Med cruise, to rarely get off the ship, is crazy talk. Lighter port days after long days is one thing. Planning to skip most days? No way. Unless you have unlimited money and time, in which case, I'm all for it.