[Flight] Refactor emit and process functions (#27234)

Since we no longer have externally configured "process" methods, I just
inlined all of those.

The main thing in this refactor is that I just inlined all the error
branches into just `emitErrorChunk`. I'm not sure why it was split up an
repeated before but this seems simpler. I need it since I'm going to be
doing similar copies of this.

Author: sebmarkbage