To argue that Birth of Tragedy has a central claim; "theory is created at the expense of art", is to attempt to reduce its offerings to an oversimplification of a single thematic component. A central claim is the result of a complete and settled debate within defined parameters. This is true even if this settlement subsequently proves merely temporary. To be central is to exist at a single point between the outer parameters of phenomena which must be known or at least defined and agreed upon. There is no single central claim that could conceivably be presented as a reference point offering access to anything like a comprehensive insight to all that this work contains. That " theory is created at the expense of art " is certainly a clearly discernable assertion made by Nietzsche. That said there are a myriad of other premises and claims within Birth of Tragedy that cannot simply be ignored.
Moreover, our difficulties are compounded when we see that some of these claims even appear to contradict each other. How then might we progress in arriving at the "kernel of this strange and inaccessible book" by this "brooding lover of puzzles" ? One possible mode of discernment is offered by the author himself via a comment he makes on another great work on which so many have differing views and approaches; Hamlet:
We find the same thing in Shakespeare, whose Hamlet, for example likewise speaks more superficially than he acts, so the lesson of Hamlet, which we have already mentioned, is to be found not in the words themselves but in a profound contemplation and survey of the play as a whole"
To gain access to Birth of Tragedy, to define its claims and to locate their relationship we must not, of course, ignore...