That's actually mostly the truth.
It's complicated, but the mayor of Las Vegas is a largely ceremonial position. It has a strong city manager government, so the mayor is just one person on a seven-person city council with no special authority beyond that. The mayor is essentially the "cheerleader" for Las Vegas, a person to schmooze at this event or that photo op.
So in addition to that, all of the casinos on the Las Vegas Strip aren't in Las Vegas anyway. They're in unincorporated Clark County, which is a different government entity (vastly more powerful than the Las Vegas City Council) and the one who calls the shots on public money for $2 billion stadiums and runs the land The Strip is on. And then there's the governor of Nevada, of course.
So if nothing else, if you're worried about me or the city of Las Vegas, don't. They're insulated by at least three degrees of separation from the mayor of Las Vegas.