at the age of の使い方

違和感なし。

He moved to Tokyo at the age of sixteen.
They married at the age of 70.
She retired at the young age of 45 and began traveling the world.

他の似ている表現と言えば、

The age of consent is 16 in the US.
We live in an age of rapid technological advancement.
By the age of three, the male lion sets out on its own.
No one under the age of 21 will be admitted.
等