After gaining experience doing both options, my opinion is the whole, "the ship will leave without you" thing is a marketing ploy that is overblown by the cruise lines to sell lower-quality excursions for more money. We bought into it the first cruise, but aren't concerned about it anymore.
With careful planning ahead, including using reputable third-parties, you are going to get back to the ship on time 99.99% of the time, and if you don't, most adults should be able to handle traveling spontaneously to catch up to the ship or head home. Yes, people have been late coming back, but it's almost always due to being careless or due to an unusual emergency that could still affect a ship-sponsored excursion (which is still third parties). In my experience, ships do actually wait a bit for those on non-sponsored excursions too. I would never count on it, because not only is it not guaranteed, but it is unbelievably rude to everyone onboard. And if they don't wait, you are on the hook for the added travel expenses, unlike using sponsored tours.
But, we have never even cut it close after dozens of private excursions. We get back with several hours to spare even (which is actually more time than the average
DCL excursion gives you in port, since most of their excursions are half day or less). If I was doing unusually high-adventure, in a place very prone to weather disruptions, or it was a very long excursions from the port, like
@Dug720 's example, I would stick to ship-sponsored.
As for which we prefer, we mix it up, depending on the port. In Europe, the DCL excursions are mostly absolute garbage (in our experience) compared to what you get through good third parties. There are a few exceptions, but you should research carefully and read many reviews. It's a bit more mixed in the Caribbean, so we look at each port and sometimes do DCL and sometimes do third-parties or self exploration.
P.S. It's one of the reasons we take our passports with us - even though DCL will tell you to leave it in your safe. I just don't trust DCL to get it to someone at the port who will get it to me if they leave without me. When traveling by land in a foreign country, you should have it on you at all times, so why would a cruise port be different? I know others here disagree, but I don't understand the thinking on why being in a foreign country from a cruise ship should be different than traveling on our own. We also have photos of the passports in the cloud, which we could access or have a family member send as backup if we needed to visit an embassy for help.
Edit: The one time we started to worry we would have a problem was when we took a private guided tour on the Amalfi coast. The guide's English wasn't great, and our Italian even worse. He kept expanding the tour beyond what we had requested, and we didn't understand that he was taking a quicker way home. I just kept reminding myself that we could easily get to Rome on our own if we missed the ship and deal with not having our luggage. It turned out that the guide knew what he was doing and we were back three hours earlier than needed.
