Birds are fairly light sleepers by nature. In the wild, they must always be on the lookout for predators that might want to make them their next meal, so it pays for them to be as alert as possible and to awaken at the slightest noise or vibration. Because of this, birds actually require more sleep in terms of hours than many other types of animals. In fact, most bird experts will tell you that the majority of captive pet birds need an average of 12 hours of sleep each night in order to maintain optimum health. That's a lot more than "40 winks!"
Restful sleep ensures that your pet birds, who are extremely intelligent and mentally active, are getting a chance for their bodies' cells to regenerate, rebuild, and repair themselves. This includes everything from your bird's brain cells down to the cells that make up his or her bones and muscles. Failure to get adequate sleep can potentially lower your bird's immune system, lead to undue stress, and cause various behavior problems. That said, making sure that your pets are getting enough shut-eye can pay off in various respects.
If you are concerned that your birds aren't sleeping enough, in spite of your avian vet's diagnosis, then it may not hurt for you to get a second opinion from a different vet in your area. The foundation of a positive ownership experience is finding a good vet that you can trust, and it seems to me that your gut instincts might be telling you that this vet isn't the right one for you and your pets.
To help your bird get more sleep in the meantime, you might want to try covering the birdcage at night. This can help provide your birds with a sense of safety and security, quieten them as it provides darkness, and can help you establish a bedtime routine with your new feathered friends. It works well for most birds and their owners, and will likely give both you and your little pets a better night's sleep if you give it a try. Good luck!