It does depend a bit on the design of the double. A good double is faster than a good single.
When my wife and I paddle our Night Heron Double, she enjoys the fact that she only has to paddle about half the time for us to keep up with the rest of our group.
When she paddles her own single, because it is a fairly efficient design, she enjoys the independence and being able to easily keep up with the rest of the group.
I understand the independence thing, but I suspect that often doubles are given a bad name because some of them are slow tubs that are not much longer than a single and wide enough to be a pain to paddle.
A double can be very useful for helping a paddler who doesn't have the strength, technique or confidence to paddle by themselves. On outings with a very leisurely group of paddlers, I have often had my front paddler swap with someone in a single who was having problems with sore shoulders or tiredness. I still remember a woman who had spent all day at the back of the group struggling to keep up. When she joined me in the double, within half an hour we had gone from the back of the group to the front. She repeatedly said that she was not competitive and didn't mind being way behind the group, but I did notice the smile on her face as we passed her husband as she pointed out to him that we had passed everyone in the group.
Our group is not competitive (or at least everyone else isn't), but sometimes it is nice to be able to keep up and still have the energy to carry on a conversation and admire the wildlife and the scenery. A good double makes that easy.
Doubles do have their uses.