I took my 11-year-old son and my 7-year-old daughter up Half Dome (with my MD wife along) and they were in low-cut "running" shoes. That's what we do all our trail hiking in – that, or low-cut "hikers" which I distinguish as having a stiffer sole than a trail running shoe.

I find the salesperson's concerns about developing feet very questionable. I'd be much more concerned about the shoes they spend 14 x 7 x 52 = 5,096 hours in each year, than the shoes they will wear on a 10-hour hike.
I've had good luck with Sportiva trail runners having good traction on both granite and snow.
Do you have your permit? If not, Cloud's Rest makes a great trip, too and no permit is required. You look down on Half Dome from there.

If doing Half Dome, my 5 biggest tips are:
1) Leave Early!
2) Like before 7 am. 6 am is even better.
3) Get to bed early the night be you LEAVE EARLY.
4) 5 am is not too early.
5) Learn where the spring is and you only need a liter of water per person at most. IF YOU LEAVE EARLY.
If you leave early, you stay cool, all during the ascent, as you gain elevation. If you start (too late) in the warmth, it will only get warmer and sunnier all day long. You also avoid most of the crowds by leaving early. And you avoid any chance of afternoon thunderstorms which is the where most of the hazards are (lightning strikes, slipping on wet rocks, etc).