Perhaps I have an odd perspective on Her Majesty but I'll post it anyway.
My Mother was raised a Japanese aristocrat. In some ways, the Japanese were very much like the British. When my brothers and I were growing up, my Mother was never one for open displays of affection. She and I used to fight about it all the time when I was a teenager. I had the comparison of my father who was very open about displaying affection to us kids and to Mom as well.
Three months after my older son was born (and UI had a niece and nephew that were 11 months old), my oldest brother committed suicide. That event completely changed my Mom. It's a hell of a way to learn but in the aftermath of his death, she learned to value open, physical displays of affection such as hugging. As a result, my sons and the other grand-kids never felt the same lack of affection from my Mom that my brothers and I did when we were growing up. They never knew the doubt of whether she loved them because she was not typically American in her demonstration of affection.
I think the Queen had the same type of restraint when it came to showing affection. I also think that as times changed, she was able to change some along with it and become more openly affectionate. She will probably never be what one could call "doting" in public but if you are accustomed to dealing with a more restrained type of person, you can see all through her married life that she had genuine and deep feelings for both her children and her grandchildren. Naturally, being affectionate comes easier the longer you do it and I think that's a lot of what we are fortunate enough to get a peek into.
***One of my favorite "Granny Lillibet" videos that I have ever seen is on YouTube. It is a short footage taken through a window showing Her Majesty and Prince William dancing together in a hallway or some such. They are dancing modern dance and you get glimpses of HM's face. She is having a wonderful time. If anyone wants, I'll dig it up and post it.***
I am also going to take the opportunity to say something that has stuck in my craw for years. The so-called "cold speech" she gave after Diana's death....if you are familiar with that sort of restrained personality, you could see a ton of emotion in her face and voice. You could see and hear how much she was personally grieving and holding up for the sake of the Princes. I think she acted correctly in a nearly untenable position of choosing between her grandsons' needs and the desires of her subjects.
The one thing I have always gotten from her appearances is how deeply she loves and cares. I'm glad that she has lived long enough to be able to express her feelings more freely in private, if not always in public.