Come to think of it--the object in Kane's childhood wasnt really the issue--it was his lost childhood that mattered--maybe he did talk about that to people he knew but they didnt clue in(he hinted about it to his second wife when discussing things moved from his homestead).
He may well have only remembered the name Rosebud when he held the snowglobe.
The rebel base exposure is more of a problem since it goes against the character. Leia wanted to protect the base more than Kane wanted to talk about his childhood.