While I do not feel there are any disadvantages or risk to routine blood testing, there is one exception which I will mention in a moment. The concerns I hear from people are
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the testing is too stressful
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the procedure is dangerous because birds are not able to spare the blood
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I know a vet that never does blood tests on birds
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expense
Yes, I feel stressed every year when I see my internist and he orders to have my blood drawn. I do not have any doubt there is a degree of stress present. However, your bird is able to deal with the stress easier when he is healthy or has subclinical disease than when the bird is severely sick.
I read online folks that say their bird died due to the loss of blood when blood was drawn from their bird. In this day and age, veterinarians that handle birds routinely have equipment that will give results on very small volumes of blood. If they don't have the equipment, there are speciality laboratories that may perform the test for them. On a finch, you are able to draw one-tenth of a cc without harming the bird. With this one tenth of cc I am able to perform a chemistry profile and generally a blood smear letting me evaluate the red and white blood cells. That is a lot of information from a extremely small sample.
There are veterinarians who see birds ocassionally that do not draw blood. However, any veterinarian that is doing routnine bird work is not able to provide quality care without blood work.
Generally, when expense is the factor it means the caretaker does not preceive a value to the testing. The statistics show routine testing will increase the quality and length of your bird's life. If your bird is an important part of the family, you need to be performing diagnostic testing. If you would spend a wad of money on a sick bird, you would spend little money before your bird has suffered quietly until it is seriously ill. When it comes to birds, Momma was definitely right: An ounce of prevention is worth a ton of cure. Without diagnostic workups, we are unable to identify disease in the early stages.
Exception: A rare complicaton of handling budgies is cardiac racing sydrome which occurs when the bird is held held for examination or any procedure. The heart of these birds race up to 1000 beats per minute and they die within about 2 seconds after it occurs without any warning. This unusual situation is rare, but it does occur. I have had this happen to me with one budgie. It is a horrible feeling when it occcurs. However, I have been able to identfy and help a lot of other budgies with the use of lab work. When I look at the entire population, the benefits outweigh the risk.