I would buy this argument until they advertise it. Everyone knows what the lottery is, there is basically zero reason to do any advertising for it.

Advertising it is encouraging addictive behavior.

I think it is fine that the state runs it, but not the way that many do.

It being a charity doesn't absolve them from bad behavior, plenty of charities have been... controversial.

A massively profitable charity towards public works funded exclusively by poor and lower-middle working people who are addicted to waiting for their numbers each week

Well, there is a fine line betweeen entertainment and a drug. That is precisely what makes entertainment so protected by modern governments and corporations: they know where that line is, and how to cross it.