Christina, that phrasing is extremely common. I prefer it to what you have suggested. I also prefer “age” to “aged.” And, if I used “aged,” I would actually write “aged 16 years.” To say someone has aged 16 sounds off unless you have previously mentioned years. I would compare it with the way we traditionally form adverbial constructions with time and the like. “She studied three years.” “She grew 3 inches.” “She is 3 inches taller.” You could omit the noun in all three, but it would seem odd if you had not already mentioned it. This really doesn’t matter much since age is so commonly written this way, but it’s a personal preference.
“Age 16” is interesting. Having checked three dictionaries, I can say their examples don’t include this style. Interesting.
This is also common:
Harry went to the store with Sally, 16, and bought a Popsicle.
The way age is used here has always stricken me as odd, but as a member of the media, I’m accustomed to it. Traditionally, an adjective would be considered a misplaced modifier here. If I wanted to insert “happy,” for instance, I would have to add a relative clause. If we want something more comparable, we can include a height reference. I would still have to use a relative clause.