The main ones people know about in the White Mtns. are Pacheta and Chitty. I haven't been to either one. Pacheta is probably the kiddie hike he was referring to. My son went to Chitty, said it was very steep. It got burned over a few years ago, too. I doubt roads are open to it at this time of year. Also, the HAZ write-up doesn't make it sound at all appealing.
Another kiddie hike is the waterfall at the G&F White Mtn. Grasslands area. Nice waterfall, but only runs during spring irrigation. (See my photos.)
There is said to be a very nice waterfall in a side canyon leading into Bear Wallow Creek, hidden way up in there, close to the reservation boundary. I had the directions at one time. Hike to rez boundary, turn around, find (first or second?) side canyon to the right as you are heading back upstream... That is NOT a kiddie hike. I can find out where it is, but I would have to wait until late May or early June to go looking for it. I tried to find it one time but was with a group of folks who didn't like thick bushwhacking.
There are some smallish but pretty waterfalls in the East Fork of the Little Colorado River. Some are basically along the East Baldy trail. Others are on the downside of 273, hike down along the creek from 273 toward Greer.
Likely there are others in the Greer area, but ain't nobody sayin'. There are a lot of creeks (Halls, Benny, Rosey, etc.) coming down through the Greer area, some with fairly steep dropoffs.
I am told that below Grant Creek cabin in the Blue Range, below the Moonshine Park trail, if you follow down the creek, there is a waterfall down in there. Haven't seen it. In fact, there are likely other falls along those creeks in the Blue Range. KP North Fork is another area to look for them.
There is a point of no return unremarked at the time in most lives. Graham Greene The Comedians
A clean house is a sign of a misspent life.