I think part of the problem is that neither William or Kate put in the hard yards before this, so they already have a reputation for being lazy and not wanting to do much, then now he has done nothing but go to a couple of jollies in 3 months. They don't have a lack of privacy all the time. They could just stay off social media or get their staff to summarise the important stuff they need to know for them. They have months off for holidays a year, when we don't know where they are and they are not disturbed, and deals with the press that rightly keep the children completely protected unless they decide to wheel them out. They have holidayed all over the world and we don't get to know about it until they release a Christmas card or photo, or they turn up suspiciously tanned from a month in Sandringham.
Being King is not nearly as onerous as being a Head of State with actual responsibility. And in many cases, they have weeks or months to prepare for that role. The only benefit of having an hereditary Monarchy is continuity, stability and the fact that the person who is destined to be Monarch more or less can be prepared for that role from birth. If William can't cope with that minimal responsibility, despite knowing that he would be doing it from birth, then maybe he should be the one to end the Monarchy and give the country a few decades to think of an alternative. The Queen was a young woman, grieving for her father, who didnt expect to be Queen until she was an older child, with two tiny children and a reluctant husband. He is a middle aged man with a compliant wife, a living father, grandparents that lived until they were nearly 100 years old.