That's too literal of interpretation imo. It shouldn't be just simply craving, desire or attachment. It should be interpreted as obsession, excessive craving, desire or attachment. Monks have family and friends too. Dalai Lama eats too which stems from the craving for food, baths too which stems from his desire to be clean and is fond of his family and friends and his fellow Tibetans, which stems from his attachment to those he cares about. So do those all make him suffer? I do not believe so.
And it's not just simply the excessive, obsessive and relentless craving, desiring or attaching to something that leads to suffering, it's the excessive, obsessive and relentless craving, desiring or attaching to something but not getting satisfied that leads to suffering. If all the excessive, obsessive and relentless craving, desiring or attaching to something all get satisfied at the end, then there is no suffering. If every single time Bill Clinton (who has sex addiction) obsesses over the sexual desire and gets that sexual desire satisfied, then there is no suffering for him. It's when he doesn't get what wants, that's when he suffers.