The reasons for royal and aristocratic children being sent to their own establishments at very young ages were a mixture of pragmatism and tradition.
It's worth noting that many foreigner visitors to England did think it was odd that aristocratic children were habitually sent to other households to finish their education. In England, there was a school of thought that held parents would spoil their own children because they naturally loved them too much and that this would, literally, spoil the child's education. So, a host family was sometimes considered better for the child's long-term development and education. It also offered families, and the child, to establish a network of connections at an early age which would help them later in life.
For royal children, the motivation was slightly different. The main reason for royal infants having their own households was health-related. Contemporaries could easily see that so many children were felled by infant diseases, which they attributed to noxious smells. Since cities, where most major palaces were, carried more smells and more diseases, shipping royal children off to a separate household in the countryside was done, quite literally, for their survival. Meanwhile, their parents, of course, tended to remain in busier environments to fulfil their governmental or ceremonial duties.
There was also an element of practice needed, because one day royal children would be in charge of vast households of their own, so to expose them to that at an early age was considered beneficial for their adult lives.
If you're looking for an in depth discussion of this, I can recommend From Childhood to Chivalry: The Education of the English Kings and Aristocracy 1066-1530 by Nicholas Orme.
Claire did some Claire Chats videos on bringing up children in Tudor times and also education:
- Education in Tudor Times Part 1
- Education in Tudor Times Part 2
- Bringing up children in the medieval and Tudor periods