Given an AST, what would be the reason behind making a Walker class that walks over the tree and does the output, as opposed to giving each Node class a compile() method and having it responsible for its own output?
Here are some examples:
Doctrine 2 (an ORM) uses a SQLWalker to walk over an AST and generate SQL from nodes.
Twig (a templating language) has the nodes output their own code (this is an if statement node).
Mostly because of old literature and available tools. Experimenting with both methods you can easily find that AST traversal produces very slow and convoluted code. Moreover, code separated from immediate syntax doesn't resemble it anymore. It's very much like supporting two synchronized code bases, which is always a bad idea. Debugging, maintenance become difficult.
Of course, it can be also difficult to process semantics on the nodes unless you have a well designed state machine. In fact you are never worse than having to traverse AST after the fact, because it's just one particular case of processing semantics on nodes.
You can often hear that AST traversal allows for implementation of multiple semantics for the same syntax. In reality you would never want that, not only because it's rarely needed, but also for performance reasons. And frankly, there is no difficulty in writing separate syntax for a different semantics. The results were always better when both designed together.
And finally, in every non-trivial task, get syntax parsed is the easiest part, getting semantics correct and process actions fast is a challenge. Focusing on AST is approaching the task backwards.