I can absolutely see what you're saying, but isn't it also entirely possible that, while W and C aren't fulfilling your expectations now, they will do so in the future?
This is entirely possible, and if that comes to happen I will gladly admit that I was wrong.
That said, to me right now it seems like William is being reluctant to be a royal. I think when he's forced to he'll step up and do more, but even then I think he'll do so reluctantly. To me it often seems like when he's involved in an engagement that doesn't interest him or interacting with people that don't interest him he does so with an open disinterest. He hasn't figured out yet how to put on a face and make it seem like he's somewhere he wants to be. He doesn't make the effort.
In contrast, the Queen, DoE, Charles, and when she was alive Diana all have (or had) the ability to feign interest. To make the charity or event or whatever seem important. They make an effort, not just in sheer number of duties but also in how they perform the duties themselves. Harry also has this touch, while William doesn't seem to. This lack of apparent interest, coupled with his reluctance to be a full time royal and his desire to be, or seem, normal, makes me think that William is always going to fall short of expectations, and as long as Kate toes the line in this regards she'll always fall short as well. To me this isn't a critique of Kate at all, nor is it her failure, but William's.
Also, isn't it possible that the Queen is allowing this because this kind of time with her young family is something she would have liked to have enjoyed, too, had it now been for the premature death of her father?
I don't think the Queen's decided to give him this much freedom of her own initiative because it runs contrary to how she treated Charles and really a lot of what she's done with William when he was younger. The Queen was given a fair amount of freedom to live her life and be a mother when she first married - she just married a lot younger than William. It was only when her father's health began to fade that she began to step things up.
Likewise, Charles was given time to grow into being a full time royal. He went to post secondary, and he had a military career. It wasn't until the end of his 20s that he was pushed into leaving the military and entering into full time duties, and then into marriage.
William was granted a lot of freedom during his schooling given his position. He wasn't harassed at school by the press. He was given the opportunity to go to the post secondary school of his choosing, and to take the gap year of his choosing. He was able to have a military career, even if he was denied the opportunity to serve in some capacities - which has happened with every heir apparent and heir apparent to the heir apparent to serve in the military during wartime in recent history. He got to have a lengthy pre-marriage relationship with Kate and married on his terms. But now that he's 4 years older than his father was when Charles began undertaking more full time duties and 7 years older than his grandmother was when she became Queen I have a hard time believing that William isn't undertaking full time duties simply because they want him to enjoy his youth in a manner that they couldn't. William's not a young man anymore, he's an adult. Furthermore, being a full time royal probably isn't going to be anymore taxing on his marriage or his relationship with his son than him having a career will. They're both time consuming things, and really being a pilot has the potential to take him away from the home far more than being a full time royal ever will, especially since the bulk of his traveling as a royal can involve his wife and son coming with him.
Just like Catherine is not just "any" mother, hers and William's situation is not like any other royal couple, either. They're still in the 3rd generation and could be waiting for the throne for the next 20-40 years. I guess my main point is: why begrudge them this time before the royal maelstrom?
I don't believe for an instant that this opportunity is giving them any more time to be a close family. I don't believe for an instant that they will somehow be more together while William is a part time royal working in the air ambulance service than they would be if they were both full time royals.
This opportunity is giving William time to avoid the job he was born into. It's giving him the opportunity to play at being normal. In a lot of ways I don't begrudge him that. I don't believe that he should ever be denied aspects of his life that are private, or time where he can live in relative freedom and do what he wants. If he was to take up full time duties but still wanted to find time to sometimes fly a helicopter on the side I wouldn't begrudge him at all - that's something he's passionate about and we all should be allowed to enjoy things we're passionate about.
That said, William isn't a child anymore, nor is he a young man still trying to find his way in the world. He's an adult who comes across as trying to avoid the responsibility of his future. While William is in an extremely unique position - he is the only adult heir apparent to a heir apparent in Europe, and the only one to have lived in a hundred years in Britain - we no longer live in a world where the monarchy is so secure that the heir apparent to the heir apparent can spend his time licking stamps and shooting pheasants. William lives in a world where regardless of what he does the monarchy is under constant scrutiny. He is viewed in a way where people almost feel as though they've made an investment in him - their tax dollars are seen as supporting him and really creating the world that he lives in, however inaccurate that may or may not be - and they're not getting much in return.
I begrudge William this time because I think he's an adult who's avoiding his responsibilities, and I think a part of being an adult is taking care of your responsibilities. I think he can seriously endanger the monarchy if he continues to avoid these responsibilities because he is the future of the monarchy and he comes off as reluctant about it - and why should we, his subjects, want him if he disdains the job? I begrudge him this time because I think that even if he has to wait another 40 years before he has the top job he would be better spent taking the time now to lay the foundations for something comparable to what his father and grandfather have achieved. In as much as William has his foundation I don't think it's anything compared to what the Duke of Edinburgh or Prince of Wales have achieved, and I don't think it ever will be if William continues to be the reluctant royal. I begrudge him this time because I think that in avoiding taking a serious royal role William is conveying an attitude that he's better than being the ceremonial ribbon cutter, ignoring the ways in which members of his family have been much more than that and insulting those who have only been that.