LAS VEGAS — Betting on baseball got Pete Rose banned from the game, never to return while he was alive.But, perhaps appropriately, he found a home in the nation’s gambling capital — where acceptance wasn’t a problem.In choosing to live in Las Vegas, before the sport’s career hits leader died Monday at 83, the man known as Charlie Hustle for his relentless play on the basepaths picked a city that embraced legalized sports betting long before most of the rest of the nation caught on.Rose was a regular at autograph sessions throughout the Las Vegas area, engaging with those who approached him to talk about his glory days with the Cincinnati Reds and Philadelphia Phillies and maybe even to commiserate about why he wasn’t eligible for the Hall of Fame.“He was very patient with his time,” said Jay Kornegay, executive vice president of race and sports operations at Westgate Las Vegas. “He wasn’t in and out. After the sessions, he stayed around and spent time with the fans and answered a lot of questions. His whole history speaks for itself, but as far as the Hall of Fame and betting on sports, he was more popular than ever.”According to the medical examiner, Rose had hypertension and heart disease as well as diabetes, but his “manner of death was natural.”Rose appeared to be a certain Hall of Famer after a brilliant playing career spent mostly with h …