Of all of our cruises we've only booked a handful of times with the(any) cruise line. The only time we did was if the excursion was not offered otherwise. Our biggest reason for booking on our own is not necessarily the cost but just the freedom to do what we want, go where we want, etc. One memory is our excursion in Costa Maya (with Native Choice - excellent!). We left before the Disney buses and upon arrival went in a different direction than the cruisers. We ran into them a few times - including when we were leaving. There were people all over that were holding up the bus - shopping, using the restroom, etc. Waiting for people in that way is not how I want to spend my holiday. We recently did the stingray excursion on Castaway Cay (obviously through
DCL). It was packed - and really such a poor experience. We've had more interaction in a touch tank at the local aquarium.
On the other hand one of the best excursions we ever did was a Sea Turtle Release in Cozumel through RCI. There were about 20 people and we were split into smaller groups on the beach. We only went with RCI because at the time it was not offered any other way. I don't believe there is a Disney difference but of course others will disagree. As far as those who will tell you that the ship will leave you behind if you're not back in time...well they are correct. They ship will leave you. However, the companies running private excursions are well aware what it would mean to their business if any of their guests missed a ship. (That would sweep through social media very quickly.) Could there be an emergency? Of course, but hundreds of thousands of people do private excursions each year. I don't believe that a Disney excursion is less stressful at all. Any good company is going to create a great experience for you - they depend on word of mouth to stay in business. What ports will you be visiting?