Anmer was always a weekend or holiday retreat for them, probably never intended for a permanent stay.
The decision to move to Windsor close to the Queen , away from London, has probably to do with the slow but constant problems of her age, not being able to fulfill her duties anymore. It is still a wonder that she is around at the age of 96 years. I don't know any monarch who has not given up before that time. It could also have to do with Andrew being so close to the Queen and fear of his influence on her (being allegedly her favorite).
If that is the case, it should have been Charles moving to Windsor, but they probably have their reasons not to do it.
Living in Windsor it is quite normal to look for a school nearby for the children.
A question for the British members: What is the difference between public and private schools in GB? I have been reading a lot of english literature especially from the 19th and early 20th century and got the impression that public schools were more prestigious than private schools.
I always thought that private schools are for the rich and privileged. So I am a little confused about that.
The move.to Windsor I would say has a lot to do with getting the children out of London and giving them more privacy.
You have state schools which are government funded. Private schools which are private and Public schools which are a small set of private schools which are old and prestigious and some of them you can only board at.