On one hand, yes, they save some people. On the other hand, they are not on the official search grid rosters, don't tend to take directions well from the authorities, and can cause all sorts of havoc. They can make it harder to find people in need. In fact, when they don't coordinate with the authorities, they end up wasting the experts' time (who may end up covering the same ground) or, when they do coordinate, people in need end up getting missed because the random people who show up to help are not properly trained to conduct searches.
Finally, as this news article shows, these folks have a tendency to go to the media and present themselves as heroes. You very rarely see that from the official search and rescue/recovery teams.
Disaster response is extremely psychologically and emotionally stressful. You may ride a wave of adrenaline while you're out in the field, but most staff end up rotating out (by mandate) after several days to protect them. Even among professionals, there are a lot of people who end up relying on alcohol to ease their minds.
Most importantly, the dignity of the people being rescued/recovered is important. It is all about those people, not about the hero rescuers. They are literally in trauma and can be dangerous and/or in need of urgent help, medical or otherwise. It is desperately unfair to them when somebody untrained, with a "cowboy" attitude, shows up and flips the script. Or, even worse, the cowboy crashes his helicopter and there's one more person who needs to be rescued.
Not all people who randomly show up in their helicopters or John boats are like this, but enough of them are to sow chaos. And look at us all wasting time talking about this guy who feels slighted because his unprofessional help was refuted. It's the same story every time.
Let the professionals do their job and, at the very least, let them coordinate the volunteers who show up to help. It sounds like a great idea for everybody to show up in their helicopter, but trust me, it's not.