Your nervous system is more distributed than originally believed, but the degree to which this is true is still a bit uncertain despite it being a major point of study since the early 1800s. What is known is that the human spine contains the majority of your body's reflexive functions and motor memory effectively making your spinal cord part of a sort of distributed nervous system like you are asking about which could in theory be pushed to a greater extreme by moving more functionality into your spine. But this may not actually be as good as you would think.
Your spine is a logical nexus for performing practiced or reflexive actions because it is so much faster than having to think through the whole process.
That said, biology often prefers the "brain" model because the closer you put neural structures, the faster they can communicate. So, by clumping together all the stuff responsible for registering senses, interpreting them, and making decisions based on them, you can think much faster, and do a much better job of letting the inputs of your different senses work together to create a unified understanding of your environment.
With all of these factors in mind with regards to distributing a bird's neurology, the answer really comes down to how important centralization is to the way a bird operates. In cases like octopus biology as Chenxi GE points out, we see organisms where coordinating the activities of each appendage with each other is far less important than being able to control all the countless ways an individual part of the body can move and react to its environment. Birds however need to be able to walk, run, and/or fly which each require rapid top-down co-ordination. If a bird moves one wing and the the other wing, or tail, or feet, etc. does not react to compensate, then the bird will tumble. If a bird sees a tree branch, every part of his body needs to precisely work together to land on it. In short, if you move stuff out into the appendages, you slow down reaction times which will probably make basic mobility much more difficult, and if you put too much into the spine, it could still coordinate certain things, but it will still likely get itself killed when it encounters a new situation where the body reacts without evaluating its context.
Also, if your concern is brain size, you may be overthinking it. Brain size is not nearly as good of an intelligence indicator as you might think. Something we often see in dog breeding is that you can make an animal's brain several times smaller without making it any dumber as long as all the same neural structures are there. It is unknown how far this limit can be pushed, but when you consider that lab mice have most of the same brain structures as a human, I'd assume the limit is much smaller than nature normally cares to go for.