MSC's private island, Ocean Cay. A first look (rendering)

Not sure they've purchased it outright so much as secured a long-lease. The duration of which won't likely be public, especially as MSC is the world's largest privately owned cruise line.

I imagine their significant investment is enough to keep them involved for at least a few decades, however!

The space has been planned as a marine reserve, so hopefully there's some additional conservation effort to share in the benefit of enhancing the island. They've certainly done a good job of replanting at any rate.

You can stay on the ship to dine, yes, but these arrangements typically extend any drinking and dining plans onto the island. There will be room for upgraded options, maybe different desserts or BBQ locations, for example, but also room to upgrade the day's activities or enjoy water sports.