Nope, still not buying that it would fool everyone. Some people sure, but not as many people as it apparently does.
Look at it this way. Bruce Wayne is DC's Charlie Sheen minus the drugs. If a genius detective vigilante trained in all practical areas appeared out of nowhere in LA beating up thugs to a pulp, what would make you think Charlie Sheen is this vigilante? On top of that, Charlie Sheen would also have concrete alibis to whenever this vigilante is spotted in another part of town.
There would be literally zero evidence that Charlie Sheen and this vigilante are the same people. Tell the cops your theory and they will do two things:
1) They'll laugh in your face.
2) They will make you sit down and logically explain to you why it would be impossible based on the evidence they gathered. Examples can include Sheen being halfway across the world while this vigilante was still active/spotted in LA, Sheen being on the phone while cops are staring directly at the vigilante (Alfred can impersonate Bruce's voice), sometimes Sheen and the vigilante even being spotted in the same place at the same time, etc.
As for his parents' murder being publicly known, that goes back to the playboy act. Everyone has their own crazy way of coping with such mental scarring. In Bruce's case, everyone
thinks he acts like a stupid immature playboy
because he saw his parents get murdered - they think that's his crazy way of dealing with the loss because they never saw his
real crazy way of dealing with the loss.
Your only valid point is Batman first appearing around the same time Bruce comes back to Gotham. And even then, the first reports of a "bat man" came several months after Bruce came back.
Sorry for being so off-topic.