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Here's my theory...
Music is a human contrivance. The twelve tones used in Western music are just something we invented to make sense of chaos. Therefore, how can someone be born with an innate ability to hear pitches that don't exist in nature? In my opinion, one does develop the ability to recognize notes by their western names, but that person would not be able to then identify the other, contrived notes of Eastern music (Indian classical music uses a different set of frequencies than we do). Also, it is difficult to say anyone has perfect pitch because they can identify piano notes. Every note on the piano has overtones that accompany it. Overtones are other notes, higher than the one played, that sound in sympathy with that note. A piano, because it has 87+ strings that might ring in sympathy. A guitar, on the other hand, has only 5 other strings that can ring in sympathy, so the overtones are different. This means that an A played on piano is just different than the A on guitar...well you get the idea.
As to Beethoven, the ability to compose while deaf was not based on his ability to audiate the pitches he wrote. For one thing, he wrote his music louder and louder so he would not have to. Secondly, the ability to compose is based on knowledge of music theory, and as Beethoven proved in his early work, he was very well acquainted with the theoretical conventions of Vienna, and knew what would sound good. For one thing, people would force themselves to like anything Beethoven wrote melodically, and he simply had the know-how to take those melodies and make the orchestra work around them.
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