there are plenty of ways to avoid fraud without having to recall the identity to the public, the main being state auditing of all lotteries, which should probably happen anyways
I believe the issue with state's auditing their own lotteries is that it still leaves a fair bit of room for public corruption. By declaring winners openly, it avails anyone among the public to research potential corrupt connections between lottery winners and lottery operators.
A third party should audit the lottery, not the state. If the payouts are from a purchased annuity, this type of audit is almost certainly required - not the state auditing itself but a 3rd party auditor.