The only reason the government is involved is because organized crime is who runs lotteries in the absence of the state. People would still gamble with black market lotteries.
This isn't really a concept that makes sense. Organized crime is a state. They serve the same functions, care about the same things, and draw legitimacy from the same sources.
If it were legal normal business would do it, see for example weed or booze before and after prohibition.
I was referring to the ‘numbers game’ racket: https://en.m.wikipedia.org/wiki/Numbers_game
No - the reason they do is because they can and it rakes in tons of money. Period.