1. You might argue that auditors and other government officials would enforce integrity. However, it would be trivial to bribe them. In addition, without total transparency you couldn't have confidence in what they're saying, anyway. It's too easy for them to lie.
2. Perhaps the real winners name would be in some book somewhere, e.g. a tax log, but the real name is out of public sight. This would also be easy to rig. If it's anonymous what's stopping the lottery commission and state government from creating a fake winner?
3. Perhaps the biggest argument in favor of a pro-anonymous lottery is that you already can be anonymous by setting up a trust. This is true, but doesn't answer the question -- how do you know it's legit?
Maybe we should just cap the winnings at 10K, an amount that people wouldn't bug you so much for and change the odds such that the revenue the state gets is the same.
Then everyone wins, no? So, in this case, instead of a single winner of 580M you could've had 60 thousand people win 10K instead. Sounds pretty good. An amount high enough that you'd give it a shot for a buck, but not so high that you'd annoy them if they won.