Over the years, we have taken cruise company tours and private tours, split pretty evenly. Where private tours most often come into play is during repeat visits to the same port.
If you do a few Mediterranean cruises, you are likely to be offered the same cookie cutter highlights tours by (any) cruise line. In such cases we rent a car, do the private organised tour or use public transport.
Missing the boat on most Med cruises - especially those up and down the coast of Italy - would not be much more than an annoyance. The ports are SO close together, you nearly coukd ride a bicycle and still can sit on the pier waiting for the ship to come into the next port.
As for the famous "pier runners ", this must become a rare event. I haven't seen any "new" clips on YouTube for a long time. 😉