Not everyone impoverished by state sanctioned gambling would do so in an illegal only world. You can bicker about what the conversion ratio is but the fact of the matter is that it exists. And no, I don't know what it is.
Eventually you'll have no lawbreakers. Either because everyone's dead, or because everyone knows they'll be killed if they break the law.
Either way, you end up with a crimeless society.
Hopefully you can see the problem with "taking things to their logical conclusion."
State lotteries don't represent "state-run crime" but rather transform existing underground markets into legal, regulated alternatives. Illegal gambling operations have long existed, typically run by criminal organizations offering no consumer protections or public benefits. State lotteries provide transparency and channel revenue toward public services instead of organized crime.
Following your reasoning would suggest that legalizing and regulating alcohol or cannabis equates to "state burglary" or "state fraud" — clearly an absurd comparison. The objective isn't to convert crimes into government enterprises, but to acknowledge when prohibition is ineffective and provide a safer, regulated alternative.