In Morgoth's Ring Tolkien said:
They grow slower than mortals though their minds are faster, learning
speech before the first year. Their wills master their bodies quickly
so they learn to walk, dance, etc by their first year. Elf Children at
play would resemble fair happy children of men with little need for
governing. Their words, and mastery of their bodies would make them
seem older than they appeared in body. Might appear to be seven when
actually in their 20’s, having adult size 50 and full maturity at 100.
So it could explain, why we couldn't spot any, or why they weren't specifically mentioned in the books.
Also, to us they may appear as adults, but in body they are children. As said, they also needed little governing, they could take care of themselves.
The source also mentiones:
They don’t always live together, and yet a sundering during the
bearing of a child or its early years is so grievous to them that they
prefer to have children in good times.
From a compilation of what Tolkien said about the reproduction of Elves and children, The Facts of Elf Life:
Elvish childhood and adolescence lasted until the age of approximately
50 years. Elves tended to marry soon after coming of age, with a
one-year engagement being standard. Elves did not say they "had a
baby," they said "a baby is given to us." The most kids an elf couple
ever had were seven, the sons of Fëanor and Nerdanel. Tolkien said
absolutely nothing about elf puberty.