But perhaps on criminal charges the authorities could pursue this, because AWS would be pretty good information about his likely identity considering he most likely paid with a credit card. I hadn't thought of this before. This would be worth following up on if you wanted the authorities to try to track him down. Still a long shot, and you would need someone to actually have the resolve to make this happen by filing a complaint.
I wonder if this can be initiated by Amazon itself. If running a gambling site is against their TOS and against the law, shouldn't they be obligated to report this to authorities if they are made aware that someone was conducting this illegal activity on their platform?
Edit: on a second thought, this might result in players'/investors' IPs being handed over as well

Probably not. They'd cease providing services and I would think that would fulfill their obligation. And I really don't think any law enforcement agency is going to waste time going after an investor. In the States, they make it a habit to target the operators instead of the gamblers of online casinos. They restrict the gamblers by forcing credit card companies not to do business with internet casinos, so there's never been a reason to target individual gamblers. Much more work for a small crime than is it's worth. I don't see the fact that this is bitcoin changing that for gamblers.