I would not take a private tour to Capri (from a ship) for reasons you already identified.
I've been there twice, once on a private tour (not from a ship) and the ferries were fine. The other time was from an Oceania sister ship, so probably the same tour venders. It was a madhouse at the end, taking a small funicular to Capri harbor ...
... and I remember that one gentleman from the ship was frantic because he got separated from his wife. (It all ended okay though - he found someone he liked better.) I believe that everyone made it back to the ship on time.
Maybe skip the Blue Grotto, unless it's a bucket list item for you as it takes a lot of time. This is from the non ship tour.
Definitely go up to the top of the island - Anacapri - maybe have lunch there. On the ship tour, we had lunch at Hotel Quisiana, which is at the bottom and just okay. I remember seeing some quaint restaurants in Anacapri, though.
Explore around up at the top and make sure you see the Faraglioni Rocks!