Now for the response to this. You must not travel much or have never flown with children. Regardless of whether their is a discount or not, you must disclose the age of all children. Rules vary on the age that they want them spelled out but any 12 and under will always have to be disclosed.
On the contrary, I travel with children quite a lot, both domestic in the US and internationally. I disclosed their ages and mentioned the specific seating needs for carseats when they were little, but I don't disclose their ages now on domestic US flights, at least not when they have a parent with them. The airline would know due to the FF accounts if they cross-checked, but I don't think they routinely cross-check age, only status. I just list them as adults, and in hundreds of domestic flights over the past decade or so I've never been called on it.
It really never occurred to me that anyone would bother to state a child's age on a domestic reservation if the fare is not age-restricted. There is no value in it, because they are not required to seat you together anyway, just in the same cabin. (The FA who said that you had to pay the unaccompanied fee if they didn't sit with you was almost surely mistaken, unless you were sitting in F or in an exit row. I've been seated away from my kids dozens of times in situations where the airline changed our seat assignments after the fact, or when there was a last-minute equipment change. I'm sure that they don't want you to ditch your kids because they are concerned about their behaviour, but IME, as long as you are in the same cabin, the fee does not apply.)