The following answer that does not meet my requirements: "Because
James Gosling didn't want to."
That's the right answer, though. The language design team (Gosling, Sheridan, Naughton, later Bill Joy, Ken Arnold, etc.) decided headers caused more problems than they solved. So they designed them out, and demonstrated that they could create a perfectly useful language without the need for them.
From Section 2.2.1 of the Java Language Environment white paper:
Source code written in Java is simple. There is no preprocessor, no
#define and related capabilities, no typedef, and absent those features, no longer any need for header files. Instead of header
files, Java language source files provide the definitions of other
classes and their methods.
Redundant definitions, keeping files in sync, conflicting definitions, hidden definitions--none of these occur in Java, because you don't have headers. If you want to see a bare class definition, you can generate one from a .java file directly--e.g. most IDEs will show you the structure of a class in a sidebar, which amounts to the same thing.