I also get the whole gamut of tests run for the first vet visit too-- the big P's, gram stain, etc. alongside what I said above, the CBC and wellness.
The reason why my vets and most breeders as well as people who own rescues/sanctuaries I spoke to suggested getting a CBC for the first visit and to continue to do so annually is because it gives you a good base line for what is normal for your bird. A lot of the times there is a big range of what could be considered healthy in general, but for your specific bird if it consistently tends to be at the high end for any one number and suddenly drops to a number that would have been considered healthy in general, but in the context of your bird and relative to your bird is very low, a CBC is incredibly helpful in detecting problems early on, and most other tests won't catch these.