I had always assumed it was because that was Jack's choice but after re-reading Andrew Birkin's book, I'm now not so sure! It seems the decision was taken in 1906, when Arthur LD was already ill. Barrie received a letter from his friend Captain Scott mentioning there was a vacancy at Osborne Naval College, and Barrie immediately replied, recommending Jack: "from his earliest days he has seemed to all of us cut out for a sailor". At the time, it is doubtful that Arthur & Sylvia could have afforded sending their sons to Eton as they were not that well off. Having Jack's education and possibly future career settled could have been a factor in the parents' decision.
It was only after Arthur's death (and later Sylvia's) that the other boys were sent to Eton, paid for by Barrie. By then George was very happy at Eton but apparently "Jack experienced nothing but misery at Osborne" but never mentioned this to either his mother or Barrie, so he never got the option of leaving the naval college. It is quite possible Barrie would have sent him to Eton too if he had known, although Jack was never close to him.